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Printed on acid-free and chlorine free paperAll rights reserved (including those of translation into other languages). No part of this book may bereproduced in any form - by photoprinting, microfilm, or any other means - nor transmitted or translatedinto a machine language without written permission from the publishers. Registered names, trademarks, etc.used in this book, even when not specifically marked as such, are not to be considered unprotected by law.Composition: Hagedornsatz GmbH, D-68519 Viernheim. Printing: betz-druck gmbh, D-64291 Darmstadt.Bookbinding: Industrie- und Verlagsbuchbinderei Heppenheim GmbH, D-64630 Heppenheim.Printed in the Federal Republic of Germany To our wives,Gwynneth Neal-Freeman and Catherine Housecraft, for their love, patience, hard work and cajolary. Also to Satin, Index, Philby and IsisPreface

Transition metals comprise roughly half of the periodic table of elements. Theirknown chemistry occupies a rather larger fraction of non-carbon research literatureto make up an enormous subject which continues to grow at a fast rate. Noencyclopedia can encompass the century or more of achievement, let alone a singlebook. Here, we do not even begin to try. What is offered, however, is an outline ofa theoretical structure for transition-metal chemistry at an elementary level thathopefully provides a consistent viewpoint of this widely varying and fascinatingsubject. By 'elementary' we mean early-to-mid UK degree level, and essentiallynon-mathematical: we do not mean, on the other hand, unsubtle, lacking inprovocation or patronizing. It has often been asserted that the 'driving forces' of inorganic chemistry varythroughout the periodic table so that we must focus on A here but on B there. If bythis is meant that the major factors are A and B here and there, we have no quarrel.It is, however, utterly unsatisfactory for anyone coming to grips with the subjectnot to understand why A rules here and not there. We need an underlying structureand understanding if we wish more than to apply given recipes: something betweenthe recipes and the impossibility of deriving chemistry from quantum theory andfundamental particles. This is a tall order. The present offering is an attempt withinjust the transition-metal series. Although the last chapter relates to the lanthanideseries, we are mainly concerned with the first transition series only. A central theme in our approach, which we believe to be different from those ofothers, is to focus on the changing chemistry associated with higher, middle andlower oxidation state compounds. The chemical stability of radical species and open-shell Werner-type complexes, on the one hand, and the governance of the 18-electronrule, on the other, are presented as consequences of the changing nature of thevalence shell in transition-metal species of different oxidation state. A goodly part of any text on 'theoretical' inorganic chemistry necessarily includesan account of crystal- and ligand-field theories. Usually, however, these theories arepresented as a self-contained discipline. Although they have certainly providedwonderful opportunities for the exercise of group theory and physics within theinorganic chemistry syllabus, the student of chemistry can well be forgiven forwondering what they actually have to say about chemistry. It is necessary to goquite far into the purely symmetry-based aspects of crystal-field theory if only toexplain the number of bands that occur in the spectra of transition-metal species orthe gross features of their magnetic properties. And we do so in this book also, although we do not take the space to cover these matters all the way to the end of aUK bachelor course. What we do focus on particularly, though, and what is oftenVIII Preface

too lightly skipped over in many other texts, is the light thrown by the crystal- andligand-field theories upon 'chemical bonding and structure in the transition block.This is an interactive enterprise in that it is equally important to understand whyligand-field theory should 'work' any way.-It is also important - though subtle, sowe only make a start on it - to appreciate the utterly different nature of ligand-fieldtheory on the one hand, and of molecular-orbital theory on the other. In all these discussions, we separate, as best we might, the effects of the delectrons upon the bonding electrons from the effects of the bonding electrons uponthe d electrons. The latter takes us into crystal- and ligand-field theories, the formerinto the steric roles of d electrons and the geometries of transition-metal complexes.Both sides of the coin are relevant in the energetics of transition-metal chemistry,as is described in later chapters. We have agonized somewhat over the title of this book. Although it might putsome readers off, we stuck with it for it really summarizes the kernel of ourapproach. This is not a compendium of chemical syntheses or properties, but ratheran attempt to bring together in a single yet non-simplistic way many importantbonding and theoretical principles that hopefully make more sense of this wide andfascinating subject. We hope that the path we have plotted through this importantarea of inorganic chemistry will commend itself to other teachers. Our lecturecourses at Cambridge broadly follow this scheme, many of the central ideas ofwhich were first presented in an article in Coordination Chemistry Reviews (99,1990, ρ 199). One of us (M.G.) thanks Professors Bill Hatfield, Tom Meyer and their colleaguesat the University of North Carolina, Chapel Hill, NC, U.S.A. for their hospitalitywhilst much of this book was written.Contents

1.1 What is a Transition Element?

The transition elements comprise groups 3 to 12 and are found in the central regionof the standard periodic table, an example of which is reproduced on the endpaper.This group is further subdivided into those of the first row (the elements scandiumto zinc), the second row (the elements yttrium to cadmium) and the third row (theelements lanthanum to mercury). The term 'transition' arises from the elements'supposed transitional positions between the metallic elements of groups 1 and 2and the predominantly non-metallic elements of groups 13 to 18. Nevertheless, thetransition elements are also, and interchangeably, known as the transition metals inview of their typical metallic properties. The chemistry of the transition elements has been investigated for two centuries,and in the past fifty years these elements and their compounds have proved to be anearly ideal touchstone for many of the models which have been developed tounderstand structure and bonding. The elements range from the widespread to theextremely rare; iron is the fourth most abundant element (by weight) in the earth'scrust, technetium does not occur naturally. Elements such as gold and silver havebeen known in the native state since antiquity, whereas technetium was first preparedin 1937. Most of the elements exhibit a typical silvery metallic appearance, butgold and copper are unique in their reddish coloration and mercury is the onlymetal which is liquid at ambient temperatures. Compounds of the transition elementsaccount for the majority of coloured inorganic materials, and many pigments arerelatively simple derivatives of these elements; however, not all transition-elementcompounds are coloured. What are the common features that unite these elements? It is surprisingly difficultto find a single definition which satisfactorily encompasses all of the transitionelements. The elements occur at that point in the periodic table where the d orbitalsare being filled. The first row transition elements coincide with the filling of the 3d,the second row with the filling of the 4d, and the third row with the filling of the5d orbitals. We define a transition element as possessing filled or partially filledvalence d orbitals in one or more of its oxidation states. This definition excludesthe elements in groups 13 to 18. The electron configurations of the transitionelements are presented in Table 1-1. The outer configurations of the transition metals in Table 1-1 imply, and detailedspectroscopic investigations confirm, that the 3d orbitals lie at higher energies thanthe 4s orbitals. On the other hand, the configurations of the M2+ ions listed, in

Table 1-2 for example, reveal the loss of electrons from the 4s shell in preference tothe 3d, so that in these species the 4s orbitals are the higher in energy. The explanation of these facts is not difficult but is subtle. We recall that theenergies of all hydrogen orbitals belonging to the same principal quantum shell (n)are equal: the 3d, 3p and 3s hydrogen orbitals are degenerate. These orbital subsets

Table 1-2. The electronic configurations of the transition-metal ions in the divalent and triva-lent states.

lose their degeneracy, however, in many-electron atoms. Orbitals with smaller orbi-tal angular momentum quantum numbers (smaller /) possess increasing numbers ofnodes in their radial functions and are referred to as increasingly 'penetrating'.Thus, a 3s electron experiences a larger effective nuclear charge and is more tightlybound than a 3p electron; a 3p is in turn more tightly bound than a 3d. Next, werecall that the energy separations between adjacent principal quantum shells inhydrogen decrease with increasing n. Taking both factors together, we expect thatsooner or later, with respect to increasing atomic number, the more tightly boundorbital subsets of the nih principal quantum shell will be more tightly bound anddecrease in energy below the higher orbital subsets of the (n-l)th principal shell. 1.2 Complexes and Coordination Compounds 3

For neutral atoms, that cross-over begins around the start of the transition-metalseries. The balance between the 4s and 3d orbital energies is delicate, however, andother factors, not discussed so far, can reverse the general trend. One such factor isthe exchange stabilization associated with the filled and half-filled d shell. This willbe familiar from discussion of ionization energies throughout the first long row ofthe periodic table when one considers the marked discontinuities at the p3 and p6configurations; this theme is taken up in more detail in Chapter 8. Now consider the ionization process yielding the M2+ ions in the first rowtransition-metal series. The configuration adopted in the ion does not depend solelyupon the relative orbital energies of the (energetically close) 4s and 3d orbitals inthe neutral atom. It also depends upon the relative energies of the putative ions3dn~24s2 and 3i/"4s°, for example. Let us consider each in turn. Removal of electronsfrom the 3d shell relieves some electron - electron repulsion and deshields the 4sorbital somewhat: both 3d and 4s shells will be more tightly bound in an M2+ ion.Removal of electrons from the 4s shell, however, depletes the inner (sub-nodal)regions of their electron density with the result that the 3d orbitals are very muchless well shielded and become much more tightly bound. It is perfectly possible inprinciple, and actually the case in practice, that the 3d orbital energy dips downbelow that of the 4s orbital as a result.

1.2 Complexes and Coordination Compounds

The systematic investigation of the chemistry of the transition elements began in

the nineteenth century, and it rapidly became apparent that many of the compoundswere somewhat different from those with which chemists were then familiar. Therewas a clear difference between the behaviour of simple ionic compounds such assodium chloride and typical transition-element compounds such as FeCl24H2O. Itwas also obvious that the compounds did not resemble the typically covalentcompounds of organic chemistry. It was considered that many of the compoundsformed by transition metals were of a complex constitution, and they wereaccordingly known as complexes. The seminal studies on these complex compounds were conducted by AlfredWerner in an intensive period of work at the turn of the century.* A typical exampleof the problems that Werner addressed lies in the various compounds which can beobtained containing cobalt, ammonia and chlorine. Stable and chemically distinctmaterials with formulations Co(NH3)nCl3 (n = 4,5 or 6) can be isolated. The conceptsof valency and three-dimensional structure in carbon chemistry were beingdeveloped at that time, but it was apparent that the same rules could not apply to

* Alfred Werner (1866 - 1919) was awarded the Nobel prize for chemistry as a recognition ofthese studies in 1913.4 1 An Introduction to Transition-Metal Chemistry

these complex compounds. Werner's key postulate was that a given metal ion couldexert two different types of valence. The first of these related to the number ofanionic groups which was associated with the compound and was termed the primaryvalence. Thus, the three compounds discussed above all contain three chloridegroups and possess a primary valence of three. In modern terms, we would equatethe Werner primary valency with the oxidation state. The novel idea that Wernerintroduced was that of secondary valency, which referred to the number of groupsattached to a metal centre. The crucial observation was that the secondary valencecould refer to the attachment of both anionic and neutral groups to the metal centre.Werner also recognized that in the same way that a metal had one or morecharacteristic primary valences, a given metal ion also had a number of characteristicsecondary valences. He noted that the most common secondary valences were fourand six. The secondary valence related to the number of groups which were directlyattached to the metal atom in the first or inner sphere. Additional groups could beassociated less strongly with a more distant second or outer sphere. Neutral ligandscould occupy the inner but not the outer sphere. The cobalt centres in the threecompounds Co(NHs)nCIs (n = 4,5 or 6) all possess a primary valence (oxidationstate) of three, and the characteristic secondary valence for cobalt(m) is six. ThusCo(NH3)4Cl3 possesses four ammonias and two chlorides in the inner sphere and achloride in the outer sphere, Co(NH3)5Cl3 possesses five ammonias and one chloridein the inner sphere and two chlorides in the outer sphere, and Co(NHs^CIs possessessix ammonias in the inner sphere and three chlorides in the outer sphere. Chemicaland physical evidence was presented to support these contentions.* At the timeWerner developed a number of descriptions for the bonding in such compoundswhich were related to the structures of more familiar organic species. We will notbe concerned with these, but note that secondary valence is equivalent to the mo-dern term coordination number. The interactions in such compounds are now better understood, and the termcomplex now has a more specific meaning. Not all transition-metal compounds arecomplexes, but many are. The terms complex and coordination compound are nowused almost interchangably.

* Particular use was made of conductivity measurements of cobalt(m) and platinum(n) complexeswhich allowed a facile determination of the number and type of ions present in solution. Forexample, the compounds Co(NH3)^Cl3 would give a monocation and an monoanion (rc=4), adication and two monoanions (n = 5) and a trication and three monoanions (n=6) respectively.In some cases, it was also possible to distinguish chemically between inner and outer spherechloride by precipitation of the outer sphere species as AgCl. 1.4 Ligand Types

1.3 The Coordinate Bond

In a typical covalent bond, such as is found between carbon and hydrogen in

methane, each atom is considered to contribute one electron to the two-electron,two-centre bond which is formed. However, we can envisage a second type ofcovalent bond in which we still have a two-centre, two-electron bond, but whereboth of the electrons come from one of the atoms or from a molecule. This type ofbond is known variously as a coordination, a dative covalent or a donor-acceptorbond. A compound containing such bonding is known as a coordination compound.The atom (or molecule) which provides the two electrons is known as the donor.The other atom (or molecule) is known as the acceptor. The term complex is usedto describe a coordination compound in which the acceptor is a metal (usually, butnot necessarily, a transition metal) atom or ion. In those coordination compounds inwhich the acceptor is a metal atom or ion, the donor is known as a ligand (from theLatin word ligare, which means to bind). It is interactions of this nature which areresponsible for the binding of ligands to a metal ion and with which we will beconcerned for the remainder of this book. Note that this is a formal description ofthe donor-acceptor interaction between the ligand and the metal and conveys littleabout the actual electron distribution. It is in no way a comment about the 'real'electron distribution in transition-metal compounds. We will return to this topic inSection 1.8.

1.4 Ligand Types

It is probably true that almost every conceivable molecule, atom or ion could act asa ligand under some circumstance or other. However, certain types of ligands arecommonly encountered, and it is these, together with the vocabulary which theygenerate, that we introduce at this stage. The majority of ligands are either neutral or anionic. Those which coordinate toa metal ion through a single atom are described as monodentate or unidentate.Examples of such ligands which we have encountered thus far include water,ammonia and chloride. A more extensive listing of common ligands is found inTable 1-3. We stress at this point that there is no difference in kind between theinteractions of a metal centre with either neutral or anionic ligands. A number of general features in Table 1-3 is apparent. Complexes may becationic, neutral or anionic. Ligands may be simple monatomic ions, or largermolecules or ions. Many ligands are found as related neutral and anionic species(for example, water, hydroxide and oxide). Complexes may contain all of the sametype of ligand, in which case they are termed homoleptic, or they may contain avariety of ligand types, whereby they are described as heteroleptic. Some ligandssuch as nitrite or thiocyanate can coordinate to a metal ion in more than one way.This is described as ambidentate behaviour. In such cases, we commonly indicate6 1 An Introduction to Transition-Metal Chemistry

Table 1-3. Some typical monodentate ligands and representative complexes that they form.

the atom which is involved in coordination to the metal by italicizing it, as in theN-bonded thiocyanate in the ion [Cr(NH3)5(NCS)]2+. Ligands which interact with a metal ion through two or more donor atoms are ofparticular importance in coordination chemistry. The number of donor atomsinvolved is indicated by the denticity - a didentate (or bidentate) ligand interactswith metals through two donor atoms, a tridentate (or terdentate) through three, andso on. If two or more of the donor atoms are interacting with the same metal centre,the ligands are described as chelating and the complexes as chelates. It is generallyfound that there is an extra stability associated with complexes which containchelating ligands - the so-called chelate effect (this is discussed in detail in Chapter9). In Table 1-4 we list some common polydentate ligands together with theabbreviations by which they are commonly known. Once again, note that both neu-tral and anionic ligands are found, and that the range of donor atoms is great. Anew feature of these polydentate ligands is that they may contain mixtures of diffe-rent donor atoms within the same ligand. Note also that a range of cyclic ligands isknown, each of which provides a central cavity for a metal ion. The study of suchmacrocyclic or encapsulating ligands is of considerable current interest. 1.4 Ligand Types

NH HN 1,4,8,11-tetrazacyclo- N,N',N",N'" [Ni(cyclam]2+

L ) tetradecane, cyclam

NH HN

1.5 Coordination Number

The coordination number of a metal ion in a complex is defined as the number of

donor atoms bonded to the metal centre. In most cases it is simple to determine.The coordination number is six in the complex species [Fe(H2O)6J2+, [Fe(py)6]2+,[Fe(CN)6]4-, [Fe(bpy)3]2+ and [Fe(tpy)2]2+. Note that when chelating ligands areinvolved, it is the number of donor atoms and not the number of ligands whichdefines the coordination number. The coordination number is not so easily definedwhen we consider those organometallic complexes in which ligation involves^-bonding of two or more centres within the ligand to a metal. For example, in theanion [PtCl3(H2C=CH2)]", the platinum interacts equally with the two carbon atomsof the ethene ligands. Is the coordination number four or five? A special 1.6 Geometrical Types and homers 9

nomenclature has been developed to describe the types of interaction encountered

in organometallic compounds, and the concept of coordination number is probablynot particularly useful in this context. Again, remember that coordination number is equivalent to Werner's secondaryvalence.

1.6 Geometrical Types and Isomers

Coordination compounds show a wide variety of regular, and an infinite range of

irregular, geometries for the arrangement of the ligands about the metal centre.However, for the first row transition metals, a few geometries by far outweigh allof the others. The regular polyhedra upon which complexes are commonly basedare the octahedron (six coordination) and the tetrahedron (four coordination). Asignificant number of four coordinate complexes exhibit a planar geometry and inChapter 7 we rationalize the occurrence of this structural geometry. One of theconsequences of complexes adopting specific geometries is the occurrence ofisomers. We review these only briefly, and the interested reader will find moreinformation in the "suggestions for further reading" at the close of this chapter. Several different types of isomers arise in transition-metal coordinationcompounds, and these are described below. Structural isomers: These are compounds in which the isomers are related by theinterchange of ligands inside the coordination sphere for those outside it. A classicalexample of this phenomenon is observed in the compounds of formula CrCl3(H2O)6-As usually obtained from chemical suppliers, this is a green solid in which onlytwo of the chloride ions are coordinated to the metal. This is formulated[Cr(H2O)4Cl2]Cl^H2O. Solutions of this compound in water slowly turn blue-green as a coordinated chloride ion is replaced by a water molecule and the com-plex [Cr(H2O)5Cl]Cl2-H2O may be isolated. More commonly, structural isomers arerelated by the exchange of anionic ligands and counter ions, rather than neutralligands. Typical examples include the pair of complexes [Co(en)2Br2]Cl and[Co(en)2BrCl]Br. Linkage isomerism: This is a special type of structural isomerism in which thedifferences arise from a particular ligand which may coordinate to a metal ion inmore than one way. In Table 1-3 we indicated that a ligand such as thiocyanatecould bond to a metal through either the nitrogen or the sulfur atom, and the complexions [Co(NH3)5(7VCS)]2+ and [Co(NH3)5(SCN)]2+ are related as linkage isomers. Coordination isomerism: This is an interesting type of isomerism which can occurwith salts in which both the cation and the anion are complex ions. Consider thesalt [Co(bpy)3] [Fe(CN)6] containing one cobalt (m) and one iron (m) centre: coor-dination isomers of this would include [Fe(bpy)3] [Co(CN)6], [Co(bpy)2 (CN)2][Fe(bpy)(CN)4], [Fe(bpy)2(CN)2][Co(bpy)(CN)4], and [Co(bpy)3][Fe(CN)6]. Geometrical isomerism: This is an important topic which played a crucial role inthe development of coordination chemistry. Werner used the number of isomers10 1 An Introduction to Transition-Metal Chemistry

which could be isolated for a range of cobalt(m) complexes to establish the

octahedral character of the CoL6 species. A planar complex of the type [Pt(NH3)2Cl2] can exist in two forms dependingupon the relative spatial orientation of the two chloride ligands. They can be at 90°to each other to give the cis form (1.1), or at 180° to give the trans isomer (1.2). Cl NH3 I Cl Pt-Cl H3N — Pt-Cl I NH3 NH3

1.1 1.2

In six coordinate complex ions such as [Co(NH3)4Br2]+, a similar situation exists,

in which the bromine ligands adopt either a cis (1.3, 1.4) or a trans arrangement(1.5). The reader should note the identity of the cis isomers despite the differentdrawings (1.3 and 1.4). In a similar manner, complexes of the type [MX3Y3] mayadopt two structures, depending upon the relative arrangement of the three identicalgroups in the octahedron. If the three X groups are arranged about a single triangularface, then the/acia/ (or/ac) isomer (1.7) is obtained, whereas if they are arrangedin three of the four sites of the equatorial plane, the meridional (or mer) isomer(1.6) is obtained. NH3 Br Br

H 3 N, -Br H 3 N- H 3 N- :Co: :Co; H 3 N* H 3 N* H 3 N* 'NH 3

NH3 Br

trans 1.3 1.4 1.5

X-

foe 1.6 1.7

Notice the 'loose' use of the term octahedral to describe six-coordinate complexeswhich are based upon an octahedral geometry, but which, by virtue of the presenceof different ligand types, are of lower symmetry than Oh This is a common usagewhich should give rise to no difficulties. Note also how introduction of chelating 1.6 Geometrical Types and homers 11

ligands into the coordination shell may reduce the number of isomers which arepossible. Thus, although there are two isomers of [Pt(NH3)2Cl2], it is only possibleto form the cis isomer of [Pt(en)Cl2] (1.8). This is because the relative positions ofthe nitrogen donor atoms in the en ligand are dictated by the CH2CH2 linker group- the two donor atoms cannot 'stretch' to occupy trans positions. Similarly, it isonly possible to obtain the cis isomer of the cation [Co(NH3)4(en)]3+ (1.9).

NH3 Cl H2 I H3N. • Ν- Pt-Cl NH2 NH3 1.8 1.9

A final type of isomerism which we mention here also arises most commonlywhen chelating ligands are present. If a molecule possesses neither a plane nor acentre of symmetry, it is chiral. (This definition is not strictly correct, but willsuffice for most transition-metal complexes.) Chiral species may exist in two formswhich are related as mirror images. These have identical chemical and physicalproperties unless they are interacting with something else which is chiral, in whichcase they differ. That may be a chiral reagent (to give diastereomeric compounds)or polarized light. A typical example of a chiral complex is found when threechelating ligands are coordinated to an octahedral centre, as in the cation[Ru(bpy)3]2+. Two different forms of this cation, related as mirror images, arepossible (1.10 and 1.11). These may be separated by formation of salts with chiralanions, and exhibit different and opposite rotations of polarized light. Note also thatthe cation [Co(en)2Br2]+ (1.12 and 1.13) is chiral, but [Co(NH3)4Br2]+ is not.

CT^D 1.10 1.11 Φ'"Ό 1.12 1.13

Another way of drawing these isomers emphasizes the three-fold nature of thebasic octahedron rather than its four-fold properties (1.14-1.17).

1.14 1.15 1.16 1.17

12 1 An Introduction to Transition-Metal Chemistry

1.7 Oxidation State

Oxidation state is a frequently used (and indeed misused) concept which apportionscharges and electrons within complex molecules and ions. We stress that oxidationstate is a formal concept, rather than an accurate statement of the charge distributionswithin compounds. The oxidation state of a metal is defined as the formal chargewhich would be placed upon that metal in a purely ionic description. For example,the metals in the gas phase ions Mn3+ and Cu+ are assigned oxidation states of +3and +1 respectively. These are usually denoted by placing the formal oxidation statein Roman numerals in parentheses after the element name; the ions Mn3+ and Cu+are examples of manganese(m) and copper(i).

Box 1-1 Older texts often employ an alternative nomenclature in which the suffixes -ous and -ic are encountered. In general, these labels only apply to the most common oxidation states of the metals, -ic referring to the higher oxidation state and -ous to the lower. Using this nomenclature, copper(n) is referred to as cupric and copper(i) as cuprous. The system works well if there are only two common oxidation states for a metal ion, but if there are more, the scheme becomes either ambiguous or unwieldy as a variety of prefixes are added.

It is usually easy to define the oxidation state for simple compounds of thetransition metals. In the case of neutral compounds, we assign charges as if thecompound were ionic. Thus, MnCl2 is regarded as (Mn 2+ , 2Cl~} and is correctlydescribed as manganese(n) chloride. Similarly, WO3 as (W6+, 3O 2 ~} is tungsten(vi)oxide. Since ligands which bear no formal charges in an ionic formulation may beignored, [Cr(H2O)3Cl3] is a chromium(m) compound, and Ni(OH) 2 , NiBr 2 ,NiBr 2 -SH 2 O, NiBr 2 -OH 2 O and NiBr 2 -9H 2 O are all nickel(n) compounds. Theassignment of oxidation state makes no implications regarding the nature of thebonding within the molecule - all of the various hydrated forms of CrCl3 arechromium(m) compounds. Oxidation state is merely a formal scheme: there is noimplication that tungsten(vi) oxide necessarily contains W6+ ions. Furthermore,problems with the assignment of oxidation state can arise with even apparentlysimple compounds. Consider, for example, Fe3O4. If the compound were ionic, wewould have four O2 ions. In order for the entire compound to be neutral, the threeiron atoms must possess an overall charge of +8. The ensuing assignment of anoxidation state of +8/3 to each iron is not particularly meaningful. A compound ofthis type is best regarded as a mixed oxidation state oxide, (FeO + Fe2O3) orFe11Fe2111O4, in which there are both iron(n) and iron(m) centres. Cations and anions are treated in an exactly similar manner, remembering to takethe overall charge of the species into account. If only neutral ligands are present,the oxidation state of the metal ion is equal to the overall charge on the ion. Thus,[Fe(H2O)6J3+ and [Ni(NH3)6]2+ are iron(m) and nickel(n) complexes respectively. Ifcharged ligands are present, formal charges are assigned on the basis of an ionicdescription. Thus, the ion [Ni(CN)4]2" is treated as containing a cationic nickel centre 7.7 Oxidation State 13

and four anionic cyanides. Since the four cyanides give a total charge of -4, thenickel must be assigned a charge of +2 in order for the ion to possess an overallcharge of -2, and it is therefore a nickel(n) complex. Similarly, [MnO4] ~ is treatedas (Mn 7+ , 4O2"} and is a manganese(vn) compound. Once again* we stress that thisin no way implies that the ion [MnO4] ~ actually contains a Mn7+ ion. By the way,aqueous solutions of transition-metal compounds frequently contain ions such as[M(H2O)6]^+: as water is the most common solvent encountered in chemicalreactions, these species are often (but incorrectly) referred to as solutions containingM"+ ions (see Box 1-2). It is quite possible for a metal centre to possess a zero or negative oxidationstate. Thus, the species [Cr(CO)6] and [Fe(CO)4]2- are chromium(O) and iron(-2)complexes. We will see in a later chapter that it is not a coincidence that these lowformal oxidation states are associated with ligands such as carbon monoxide. Some ligands pose problems in the assignment of a formal oxidation state to ametal centre. Nitric oxide is a case in point. The ligand may be formulated as eitheranionic NO" or cationic NO+, and there follows the appropriate ambiguity inassignment of the oxidation state of the metal ion to which it is bonded. Theseproblems arise when it is not clear as to what charge is appropriate to assign to theligands in the ionic limit. We have repeatedly emphasized the formal character ofthe concept of oxidation state and turn now to a different general concept whichhelps us address the real electron distributions in compounds.

Box 1-2 It is very common for inorganic chemists to 'neglect' or 'ignore' the presence of solvent molecules coordinated to a metal centre. In some cases, this is just carelessness, or laziness, as in the description of an aqueous solution of cobalt(n) nitrate as containing Co2+ ions. Except in very concentrated solutions, the actual solution species is [Co(H2O)6J2+. In other cases, it is not always certain exactly what ligands remain coordinated to the metal ion in solution, or how many solvent molecules become coordinated. Solutions of iron(ni) chloride in water contain a mixture of complex ions containing a variety of chloride, water, hydroxide and oxide ligands. When dealing with the kinetic or thermodynamic behaviour of transition-metal systems, square brackets are used to denote concentrations of solution species. In the interests of simplicity, solvent molecules are frequently omitted (as are the square brackets around complex species). The reaction (1.1) is frequently written as equation (1.2).

[Co(H2O)6J2+ + 4Cl - = [CoCl4]2- + 6H2O (1.1)

Co2+ + 4Cl- = [CoCl4]2- (1.2)

Whilst this will be satisfactory when dealing with kinetic data in which reactions involving the solvent will not explicitly appear in the rate equations, it is not appropriate when we consider equilibrium constants. As an exercise, consider the formation of [Ni(en)3]2+ from aqueous solutions of nickel(n) chloride and en (en = H2NCH2CH2NH2); write the equations with the inclusion and the omission of the water molecules. Can you recognize the driving force for the formation of the chelate in each case?14 1 An Introduction to Transition-Metal Chemistry

1.8 Electroneutrality Principle

It was recognized early on that the formality of 'dative co valency' in coordination

compounds presents some difficulties. Many inorganic compounds are convenientlythought of as ionic salts in which there is an essentially complete charge separationbetween cationic and anionic species. Compounds of the group 1 and group 2 metalstend to be readily categorized as 'ionic'. Dissolution of a salt of a group 1 or agroup 2 metal results in the formation of solutions containing solvated cations andanions - only in concentrated solutions are there significant cation -anioninteractions beyond simple ion-pairing. However, the ionic model does not appearto be suitable for the description of the properties of many transition-metalcompounds. For example, the compound K4[Fe(CN)6] dissolves in water to givesolutions containing solvated potassium ions and the [Fe(CN)6]4" ion, rather thansolvated potassium, iron(n) and cyanide ions. The interactions between the cyanideand the iron(n) centre appear to result in longer lived species than result fromsimple electrostatic interactions of the type observed in sodium chloride. This is, ofcourse, the sort of argument which led to the development of the description ofcoordination compounds in terms of donor -acceptor interactions between theligands and the metal centre. Let us now examine the consequences of the formation of a donor - acceptorbond in a little more detail. If the donor - acceptor bond is completely covalent,then we record net transfer of one unit of charge from the donor to the acceptor asa direct consequence of the equal sharing of the electron pair between the twocentres. This result leaves a positive charge on the donor atom and a negativecharge on the acceptor atom. The limiting 'ionic' and 'covalent' descriptions of acomplex cation such as [Fe(H2O)6J3+ are shown in Fig. 1-1.

We have already commented that the 'ionic' structures are not in accord with thechemical properties of coordination compounds. However, there are also a numberof objections to the covalent description. The charge distribution is such that theiron(m) centre bears a three minus charge, whereas the oxygen atoms of the waterligands each bear a single positive charge. This would be unrealistic in view of theelectronegativities of these elements (Fe, 1.8; O, 3.5), which predict that the Fe-Obond should be polarized in the sense Fe^-O5". The problem was addressed byPauling, who recognized that, in reality, it was not appropriate to describe most 1.9 Rationalization of Complex Geometries 15

bonds as being purely 'ionic' or purely 'covalent'. He developed a description of

the bonding between a metal and its ligands which included considerable ioniccharacter in the metal - ligand bonds within a basically covalent regime. In order todetermine the amount of 'ionic' character within a given metal-ligand bond, Paulingframed his electroneutrality principle. In this, he opined that the actual distributionof charges within a molecule is such that charges on any single atom are within therange-1 to +1. We shall see how this works by reconsidering the ion [Fe(H2O)6J3+. An 'ionic'description results in a +3 charge on the metal centre, whilst a 'covalent' descriptiongives the metal a -3 charge. Now the electroneutrality principle suggests that the'ideal' charge on the metal centre is zero. That would be achieved if the iron centregains a total of three electrons from the six oxygen donor atoms; in other words, ifeach oxygen loses one half of an electron rather than the whole electron which thefully covalent model demands (1.18). Pauling describes this situation as 50%covalent (or 50% ionic).

1.18

We shall return to this topic in Chapter 9. Remember that the unequal distributionof electrons within bonds results in a continuous variation from 'covalent' to 'ionic'bonding.

1.9 Rationalization of Complex Geometries

The coordination geometries of main group compounds are generally rationalized in

terms of the Valence Shell Electron Pair Repulsion (VSEPR) or Gillespie-Nyholmmodel. The reader will recall that in this scheme, the spatial arrangement of atomsand groups about a central atom is dictated solely by the number of such groupsand by the number of stereochemically active lone pairs. The model only considersmutually repulsive interactions between the various ligands and lone pairs presentin the valence shell (or equivalently, between bond pairs and lone pairs in thevalence shell) and makes no assumptions about the nature of the bonding exceptinsofar as it is predicated upon a particular number of lone pairs. The assumption isgenerally made that all electrons in the valence shell - both lone pairs and bonding16 1 An Introduction to Transition-Metal Chemistry

pairs - are stereochemically active. This model is remarkable for both its simplicityand its general applicability. The basic method involves the totalling of the numberof atoms or groups and lone pairs associated with the central atom, and deriving astructure based upon the appropriate «-vertex polyhedron.

Box 1-3 These are the polyhedra which are used for the basic structural types: vertices polyhedron coordination geometry

3 triangle trigonal planar

If the central atom has different groups or atoms around it, or if one or more ofthe vertices of the polyhedron is occupied by a lone pair, then variations in bondangles will occur such that distorted polyhedral arrangements are obtained. In itsquantitative forms, the VSEPR model parameterizes each individual interaction andmakes very accurate predictions of the distortions which are to be expected. This model has been successfully applied to the structures of many thousands ofmain group compounds, and bond angles within a few degrees of the experimentallyobserved values are usually correctly predicted. The basic model is only concernedwith repulsive interactions between atoms and other atoms or lone pairs. Nowherein the model is any consideration given to the attractive nature of the bondingwhich is present (single, double or triple bonds) or to the ways in which the centralatom atomic orbitals must be utilized in attaining the desired geometry: once again,except for the assignment of the number of nonbonding electron pairs no as-sumptions about the bonding are made. Many texts suggest that the VSEPR modelmay not be usefully applied to transition-metal compounds. This is not so. Kepert has developed a repulsion model for the prediction and rationalization ofangular coordination geometry in transition-metal complexes at various levels ofdetail. His basic model considers the ligands to be arranged upon the surface of asphere enclosing the central metal ion. The distances between donor atoms ofchelating ligands are fixed as determined by intra-ligand bonding. This apart,Kepert's model allows for free variation in the angular geometry, that is, freemovement of all donor atoms (or chelate groups as appropriate) on the surface ofthe notional sphere, subject to a l/r" repulsive force between them. Nowhere in thebasic model is any consideration given to the nature of the metal - ligand bondingor to the steric potential of the d configuration. In application to thousands ofcompounds, the model consistently predicts angular geometries which agree withexperiment to within about 2°. The relative energies of these conformational minimadepend, of course, upon the value of η in the repulsion law invoked (n = 2, 6 and12 have all been investigated) but the angular positions of these minima are almostindependent of n. 1.10 Review of Properties of Transition-Metal Compounds 17

There is one striking group of exceptions to the otherwise almost unbroken

success of Kepert's approach. No model predicated solely upon the repulsionsbetween monodentate ligands (or between bonds) can account for the planarity ofsome four-coordinate complexes. Yet hundreds of planar d* complexes like[Ni(CN)4]2" or [PtCl4]2" are known. Clearly, Kepert's model is to be augmented andwe discuss this matter further in Chapter 7.

Box 1-4 The compound [Zn(tpy)Cl 2 ] (1.19) contains a planar tridentate ligand with nitrogen donor atoms. The geometry is often described as trigonal bipyramidal with the three 2,2':6',2"- terpyridine donor atoms occupying one equatorial and the two axial sites. Kepert's calculations actually predict a geometry that is far closer to the ideal square-based pyramid. His predictions are well confirmed by crystallographic analysis.

1.19

Thus far, we have only considered the angular geometry of complexes; variationsin bond lengths also pose challenges. For example, the gross inequality of bondlengths in [NiF6]3' and many copper(n) and chromium(m) complexes requires anexplanation. Questions of this kind are also addressed in Chapter 7.

1.10 Review of Properties of Transition-Metal Compounds

Finally, we summarize some of the properties of transition-metal compounds and

attempt to distinguish those which are characteristic of a transition-metal complexas opposed to any metal complex. Variable oxidation state - One obvious feature of transition-metal chemistry isthe occurrence of a number of characteristic oxidation states for a particular metal18 1 An Introduction to Transition-Metal Chemistry

ion. In general, these oxidation states are readily interconverted. This tendency toform a variety of oxidation states is displayed in Table 1-5. Note that the occurrence of a maximum oxidation state, corresponding to theremoval of all the valence shell electrons and the adoption of a dQ configuration,does not occur after manganese. In Chapter 9 we see how this reflects the contractionof the poorly penetrating 3d orbitals as the nuclear charge increases and it becomesprogressively more difficult to remove electrons. The exhibition of variable valency is indeed a characteristic of transition metals.Main group metal ions such as those of groups 1 or 2 exhibit a single valence state.Other main group metals may show a number of valencies (usually two) which arerelated by a change in oxidation state of two units. This is typified by the occurrenceof lead(iv) and lead(n) or thallium(m) and thallium(i). However, all the transitionmetals exhibit a range of valencies that is generally not limited in this manner. Low oxidation states - An important characteristic of transition metal chemistryis the formation of compounds with low (often zero or negative) oxidation states.This has little parallel outside the transition elements. Such complexes are frequentlyassociated with ligands like carbon monoxide or alkenes. Compounds analogous toFe(CO)5, [Ni(cod)2] (cod = 1,4-cyclooctadiene) or [Pt(PPh3J3] are very rarely en-countered outside the transition-metal block. The study of the low oxidationcompounds is included within organometallic chemistry. We comment about thenature of the bonding in such compounds in Chapter 6. Colour - A striking feature of transition-metal compounds is their colour. Whetherit is the pale blue or pink hues of copper(n) sulfate and cobalt(n) chloride, or theintense purple of potassium permanganate, these colours tend to be associated mostcommonly with transition-metal compounds. It is rare for compounds of main groupmetals to be highly coloured. Unpaired electrons and magnetism - One of the consequences of the open(incompletely filled) dn configuration of transition-metal ions may be the presenceof one or more unpaired electrons. Such compounds could be described as radicals,and they are detected by techniques such as electron spin resonance spectroscopy. 7.70 Review of Properties of Transition-Metal Compounds 19

However, while transition-metal ions often contain unpaired electrons, they exhibitnone of the reactivity that is commonly associated with such radicals outside the dblock. There is no behaviour comparable to that of the highly reactive and shortlived radicals such as CHs. Also associated with the presence of unpaired electronsin these species is the phenomenon of paramagnetism. The long - term stability ofmany compounds with unpaired electrons is a characteristic of the transition-metalseries. Formation of coordination compounds and variable coordination number ~ Boththe transition and the main group metal ions form coordination compounds. Thereis no difference in kind between the complexes formed between cobalt(m) andammonia and those between lithium and water. Though the absolute stabilities mayvary, large ranges of stability constants are observed for both main group andtransition-metal ions. Transition-metal complexes may gain or lose ligands to changegeometry and so do main group complexes. The existence of coordination chemistryin the transition-metal block does not set these metals apart from those of the maingroups.

Suggestions for further reading

- This is an easy to read introduction to the area.2. J.E. Huheey, E.A. Kieter and R.L. Kieter, Inorganic Chemistry, Harper Collins, New York, 1993. - An excellent general introduction to inorganic chemistry, with first rate chapters dealing with transition metal chemistry.3. RA. Cotton, G. Wilkinson, Advanced Inorganic Chemistry, 5th ed., Wiley, New York, 1989. - A relatively comprehensive work with a great deal of descriptive material concerned with transition metal chemistry.4. A.G. Sharpe, Inorganic Chemistry, 2nd ed., Longman, London, 1992. - A general text with a number of relevant chapters.10 The Lanthanoid Series

10.1 The Lanthanoid Contraction

Thus far, we have focused exclusively upon the d-block metals. For some, the term'transition elements' defines just these J-block species; for others, it includes therare earth or lanthanoid elements, sometimes called the 'inner transition elements'.In this chapter, we compare the J-block and /-block (lanthanoid) elements withrespect to their valence shells. In doing so, we shall underscore concepts which wehave already detailed as well as identifying both differences and similarities betweencertain aspects of 'main' and 'inner' transition-metal chemistry. We make no attemptto review lanthanoid chemistry at large. Instead our point of departure is the mostcharacteristic feature of lanthanoid chemistry: the +3 oxidation state. The lanthanoids occur under scandium and yttrium in the periodic table. Someuseful data are presented in Table 10-1. On crossing the series of +3 ions fromlanthanum to lutetium, observe the variation of electronic configuration 4/" from η= O to η = 14. Note too, a decrease in ionic radius by about 20% across the series.The reason for a size reduction across any series in the periodic table is, of course,the increase in effective nuclear charge that results from the inefficient mutualshielding of electrons in the same shell, together with the monotonic increase inreal nuclear charge. However, the phenomenon is particularly marked in thelanthanoid series and is referred to as 'the lanthanoid contraction'. Perhaps themain reason for drawing attention to it by this special name lies in its consequencesfor the chemistry of the third row d-block elements that follow the lanthanoid series.That chemistry is strongly affected, as ever, by ionic radius. The increase in size ofthe third row J-block elements relative to those of the second row, which is expectedfrom the increased number of electrons and the higher principal quantum number ofthe outer ones, is almost exactly offset by the intervening lanthanoid contraction. Inconsequence, there are more similarities between the chemistries of the second andthird row transition metals than between the first and second row elements. The magnitude of the contraction across the whole lanthanoid series is due partlyto the length of the series - the placement of up to 14 electrons in the 4/shell - andpartly to the poor shielding of one / electron for another. The poor shielding arisesin two ways. Firstly, the form of the radial wavefunction for 4/electrons, like thatof the 3d, involves no inner maxima. Thus, these / orbitals are not of the 'penetrating' type, and so little inner electron density is available to shield the outerregions from the (increasing) nuclear charge. Secondly,/orbitals are often describedas 'diffuse'. It is important to be clear about the use of this adjective here, however,

*Outside of closed [Xe] shell.

since they are not radially diffuse. Indeed, as we shall discuss at length shortly, theyare radially compact. They are angularly diffuse in that / orbitals have many morelobes than d orbitals, for example. The / orbitals have a larger angular spread andsince they are normalized to unity - corresponding to their housing exactly oneelectron (of a given spin) - the local electron density is rather low. This, too,contributes to the poor self-shielding within the 4/ shell.

10.2 The Core-Like Behaviour of/Electrons

The radial spread of the 4/orbitals in lanthanoid +3 ions is so limited that / electrondensity is almost totally confined to the inner regions of the lanthanoid ion. Figure10-1 schematically compares the radial waveforms of the 4/and 6s orbitals. Thesituation is rather like that in Fig. 2-1 for the 3d and 4s orbitals in the first row dblock for higher oxidation state species. We have seen what consequences flowfrom the relative 'isolation' or 'uncoupling' of the d electrons in such circumstances.In particular, recall how the chemical bonding between a J-block metal and itsligands is effected within a metal valence shell that is largely 4s; and how the delectrons modify that bonding by 'secondary', repulsive and other non-overlap 10.3 Magnetic Properties in the f Block 199

means. The same is true for the lanthanoids, but to a greater extent. The core-likebehaviour of the 4/ orbitals in lanthanoid +3 ions is even more pronounced thanthat of the 3d orbitals in analogous d-block complexes. There is no significantcontribution from the 4/ shell to the valence shell in lanthanoid chemistry. Thisdoes not imply an ionic chemistry of Ln3+ species, however, any more than that d-block bonding is primarily ionic. We discuss evidence of covalency in lanthanoidcomplexes below. First, we look at the consequences for the /-electron shell of its'contracted' or core-like character.

10.3 Magnetic Properties in the/Block

The number of terms arising from a given/" configuration is generally much largerthan from dn and follows directly from the greater degeneracy of the / shell. Weshall see something of the complexity of/-block term diagrams in the next section.However, it is quite simple to work out the ground terms in the / block by usingHund's rules. For the /6 configuration, for example, we maximize spin by placingeach electron in a separate orbital (so reducing the interelectron repulsion energy).

m, 1 O -1 -2 -3 T T T t t T

The total ζ component of the spin angular momentum, M5, is given by the sumΣ ms· = 3 and implies a total spin for the ensemble of S - 3 and a spin-multiplicity 6(2S + 1) of 7. Similarly, Σ m, = 3, yielding L - 3. The ground term of/ is therefore1 F. Spin-orbit coupling, which arises from the magnetic interactions amongstelectrons, splits the 1F term into levels 1F1 with / values varying from the sum (L +S ) to the difference IL -51, as usual. We find / ranging O to 6 in this case. For less-200 10 The Lanthanoid Series

than-half-filled shells, as here, Hund's third rule places the level with the minimum/ value lowest in energy; so the ground level is 7F0. Lande's interval rule definesthe energy separation between two adjacent levels as λ, the spin-orbit couplingcoefficient, times the larger of the two / values. Collecting all these results together,we establish the splitting pattern for the ground term of/ 6 as shown in Fig. 10-2a.It is left as an exercise for the reader to show that the corresponding ground termand level stacking for/ 8 ions are as shown in Fig. 10-2b. Now the magnitudes of the spin-orbit coupling coefficient are much greater inthe / block than the d. Consequently, the energy separations between levels for /-block ions are usually much larger than the ambient thermal energy, kT. For the /8ions of terbium(m), for example, λ = 270 cm"1 and the first excited level, 7F5, lies1620 cm^1 above ground. At room temperature, kT ~ 200 cm"1 so that the populationof the 7F5 level is about ^-1620/2009 which is negligible. We saw in Chapter 5 that the paramagnetism of a system primarily depends uponthe splitting of populated states within an applied magnetic field. So, for an(obviously unobtainable) sample of Tb3+ ions, the magnetic moment is primarily afunction of the 7F6 level alone. There are second-order contributions arising fromthe mixing of higher levels into the ground level by the magnetic field but, as theseare inversely proportional to the energy separation between the mixing levels, theyare generally small. An analytical formula for the effective magnetic moment, jUeff,has been derived for the case where only one level, 25+1L/, is thermally populatedand second-order contributions are ignored (Eq. 10.1).

(10.1)

In Eq. (10.1), g is the 'Lande splitting factor' and is given by the expression in Eq.(10.2). 10.3 Magnetic Properties in the f Block 201

/(J + I)-L(L + I) + W + I) 2 J(J +1)These formulae explicitly involve only the L, S and J quantum numbers that definethe ground level 2S+1L/, as required. The discussion thus far refers to free ions. One can apply the formulae to themagnetism of J-block compounds but it fails utterly to reproduce experiment. Theinapplicability of Eq. (10.1) is due to two factors: a) the smaller magnitude of λ -at least in the first row d metals - means that level splittings in the free ions aresmaller and molecules significantly populate more than just the ground level, andb) the formula takes no account of the ligand-field splitting of the free-ion terms.We have seen that such splittings in the d block are of the order of several thousandwavenumbers, a perturbation that dwarfs the effects of spin-orbit coupling. Indeed,spin-orbit coupling is manifest in the magnetic properties of first row d-blockcomplexes largely as a correction to the 'spin-only' formula (though thesecorrections are very important for ions with T ground terms). The neglect of the ligand field in Eq. (10.1) leads one to expect no satisfactoryaccount of the experimental magnetism of lanthanoid complexes either. It is anempirical fact, however, that Eq. (10.1) accounts extremely well for observedmagnetic moments in most lanthanoid compounds. We compare typical experimen-tal moments for lanthanoid complexes with those calculated from Eq. (10.1) in Fig.10-3. Significant discrepancies occur for/ 5 and/ 6 species and we will comment onthese shortly. The question therefore arises of 'why does neglect of the ligand field, implicit inEq. (10.1), not matter for the / block while it is utterly unacceptable for the dblock?'. The answer is both trivial and subtle. Trivially, the neglect is acceptable

for the lanthanoids because the ligand field itself is negligible. The / electrons aresufficiently buried beneath the valence shell so that they are affected very little bythe ligand environment. The bond orbitals engaged in binding a lanthanoid metal toits ligands largely lie beyond the radial extent of the / orbitals so that ligand-fieldsplittings are very small. We shall see in the next section that splittings of tens ofwavenumbers in the/series replace the thousands of wavenumbers in the d series. The subtle reason, which we do not detail here, is as follows. The splittingscaused by normal laboratory magnetic fields are of the order 0.1 to 1 cm"1 only.These are, of course, small even compared with the small ligand-field splittingstypical of lanthanoid complexes. One might expect, perhaps, that even these smallligand-field splittings cannot be ignored when considering magnetic properties.Actually, such expectations are not as reasonable as might first appear, partly becausemagnetism is about the changes that occur on application of an external field.Nevertheless, it requires a theorem due to Van Vleck to show generally andirrefutably that, so far as mean (spatially averaged) magnetic moments areconcerned, ligand-field splittings which are no larger than about kT have nearnegligible effects upon paramagnetism. Incidentally, this same theorem explainswhy so many of the simple formulae described in Chapter 4 work satisfactorilyeven when molecular geometries depart slightly from rigorous octahedral ortetrahedral symmetry. It is a crucial theorem for the theory of paramagnetism. Thisbook, however, is not the place to demonstrate this important result. Altogether, we can say that the success of Eq. (10.1) in reproducing the magneticmoments of lanthanoid complexes is due entirely to the very small magnitude ofthe ligand-field splittings and so, in turn, to the contracted nature of the/orbitals. We conclude this section with a further commentary on the discrepancies notedfor the/ 5 and/ 6 systems. Consider first the case of the/ 5 samarium(m) complexes.The ground term is 6H with six levels ranging / = 15/2 to 5/2. The 6H572 level islowest in energy with the first excited level lying 1λ/2 above it, or some 1650 cm"1.The second-order Zeeman effect, relating to the admixture of the first excited 6H772level into the ground 6H572 level (and, indeed, of the yet higher lying levels) is notnegligible in this case. More complete calculations which include these second-order effects, as Eq. (10.1) does not, do actually reproduce the observed moments 5 6for / species very well. Second-order Zeeman terms are also important for / 7europium(m) species. In this case, however, the ground level is F0, as we showedabove. Equation (10.1) yields a zero moment for this level. The same result can be 7arrived at as follows. The degeneracy of the F0 level is (2/+1) = O. A singlydegenerate level cannot split in a magnetic field (or any other, of course) and sogives rise to no first-order paramagnetism. The first excited level, 7F1 lies Iλ aboveground, or about 230 cm" 1 . (Note, by the way, that for/ 8 ions, the 7F5 level lies 6λ= 1380 cm-1 above the ground 7F6, so illustrating a major difference between/"configurations and their 'hole equivalents'). Accordingly, there is a significantpopulation (e-230/20°) of the first excited level for/ 6 . So, in addition to any se-cond-order Zeeman corrections, we must include first-order terms relating toelectrons populating this excited level. Once more, a full calculation of these effects 6does indeed reproduce the magnetic moments that are typically observed for /species. 10.4 Spectral Features 203

Overall, then, the magnetic moments of all lanthanoid complexes are wellreproduced without reference to the ligand field; inter alia, we can infer that theligand-field splittings in /-block complexes are no greater than about kT at roomtemperature.

10.4 Spectral Features

Part of the absorption spectrum of an aqueous solution of neodymium(m) -

configuration/ 3 - is shown in Fig. 10-4. The situation shown there is quite typicalof the whole of the lanthanoid series i.e. we could have chosen any/" configurationequally well to illustrate the main characteristics of the spectra of lanthanoidcomplexes. We shall focus on three main features: splittings, band widths and abso-lute excitation frequencies.

The solution spectrum is remarkably similar to that of the corresponding free ionin that slightly split groups of transitions replace the free-ion spectral lines on aone-to-one basis. These splittings are the ligand-field components. That theyoccasionally seem so numerous is due to the more complex geometry of the species,which is probably [Nd(H2O)9J3+, as well as to the fact that the greater degeneracy ofthe / shell relative to the d shell begets more sublevels even for the samecoordination geometry. We shall not concern ourselves with the details of thesesplitting patterns. Suffice to note that here is the direct evidence of the very smallligand-field perturbations that were deduced from the magnetic properties discussedin Section 10.3.204 10 The Lanthanoid Series

By and large, the spectral bands are very sharp as compared with 'd-d'transitions in J-block complexes. The sharpness of these '/-/ transitions followsimmediately from the core-like character of the / shell. It interacts little with thebond orbitals and other aspects of the environment - hence the small ligand-fieldstrength - and so the range of interaction throughout a molecular vibration is alsosmall. The intensities of these transitions are also small. Typical extinction coef-ficients for 'd-d' transitions in centrosymmetric complexes are of the order 5- 10.For acentric chromophores, like the tetrahedra, they may be around 500. Those inthis (typical) lanthanoid system are about 5. Considering that the coordination isprobably that of a tricapped trigonal prism, which is non-centrosymmetric, the'/-/ intensities are some two orders of magnitude less than might be expected fora similar J-electron system. Once again, we understand this result in terms of themuch smaller overlap between the metal 4/ orbitals and the bond orbitals. Themetal / - ligand orbital mixing cannot be zero, for otherwise Laporte's rule wouldensure vanishing intensities, but it is very small: smaller than for the d block and,as we saw in Chapter 4, that is small anyway. Before moving on to the absolute transition energies in lanthanoid spectra, let ustake stock. The resemblance between the transition energies of lanthanoid complexspectra and those of the corresponding free ions, taken together with the sharpnessand weakness of the bands and the small magnitudes of the ligand-field splittings,all concur with the notion of a well-buried / shell. The magnetic moments oflanthanoid complexes similarly support this view. The / electrons comprise a welldecoupled subset of electrons within these complexes, bequeathing to the metal 6s(and perhaps other) orbitals the role of the valence shell. We thus observe a situationlike that described for the d electrons in the main transition-block (in higheroxidation states) complexes, but much more obviously. Both classes of compounds,however, are covalent in that complex species retain their integrity in manyenvironments. Evidence for that covalency comes directly from our last topic, namely, the abso-lute transition energies of complex 'f-f spectra. Many of the spectral bands in thespectra of lanthanoid complexes involve transitions between levels of the sameterm and, as such, provide a measure of the strength of the spin-orbit coupling. It isgenerally observed that these interlevel spacings are smaller than in the spectra ofthe corresponding free ions. Thus, A(complex) < A(free ion). Other transitions occurbetween components of different terms. After appropriate (and, unfortunately, rathercomplicated) analysis, one may determine the magnitudes of the various interelectronrepulsion parameters, which include, for example, the Racah B parameter discussedin Chapter 6. It is found, quite generally, that 5(complex) < 5(free ion). Themagnitudes of these nephelauxetic effects are roughly of the same order as foundfor J-block complexes. Similarly, the reductions in λ values, as above, - andobtained again only after lengthy analysis - are of similar proportionate magnitudesin d- and/-block systems. Both phenomena attest to the covalency of the chemical bonding in these species.Incidentally, they also highlight the different characters and implications of thespectrochemical and nephelauxetic series. Within either lanthanoid- or (higheroxidation state) d-block species, the ligand orbitals overlap with the metal s functions 10.4 Spectral Features 205

and donate electron density to the metal. The s orbitals (4s for the first row d block;6s for the lanthanoids) are of the penetrating type (inner maxima) and that portionof the ligand electron density which occupies these inner regions is particularlyeffective in shielding the outer, 'spectral', 4/or 3d electrons from the nucleus. The/(or d) orbitals expand somewhat and so the average distance between f (d) electronsincreases and the interelectron repulsion parameters decrease. It is also the case,though we do not enlarge on the matter here, that the magnitudes of spin-orbitcoupling coefficients are inversely related to a power of the mean distance of (spectral) electrons from the nucleus. The reduction in λ values goes hand-in-handwith the reduction in B values, though not pro rata. The central point, then, is that tiny ligand-field splittings and 'normal' sizednephelauxetic effects in lanthanoid spectra are not at all contradictory. The onereveals the isolation of the/shell, the other attests to the normality of the metal-ligand bonding.

Suggestions for further reading

1. Systematics and Properties of the Lanthanides (Ed.: S.R Sinha), Reidel, 1983. - Here, the article by Hiifner shows energy levels throughout the /-block.2. J.H. van Vleck, The Theory of Electric and Magnetic Susceptibilities, Oxford University Press, Oxford, 1932. - This is a great original - see Chapter 9.3. S.A. Cotton, Lanthanides and Actiniae s, MacMillan, Basingstoke, 1991. n2 Focus on the d Configuration

2.1 Spectral Features

We discover a far-reaching generality of transition-metal compounds simply by

looking at bottles on the laboratory shelves. By and large - and with manyexceptions to be sure - compounds of transition metals are coloured, while those ofthe main group metals are not. Furthermore, the colours are gentle rather than vivid- weak rather than strong - and often group together with the metal ion involved.Thus, many copper(n) complexes are blue, while those of nickel(n) are green;manganese(n) compounds are only weakly coloured; a wide range of colours areassociated with the different oxidation states of vanadium. Look again and thesegeneralizations are seen to fail, but there are clearly some patterns to be found. Weshall expend considerable time and effort discovering and understanding thesepatterns and generalities, not just because it might be fun to make theories aboutthe pretty colours but because they are the outward manifestations of much of theunderlying electronic structure in transition-metal complexes. To be honest, it isonly with hindsight that we can say what is probed by the spectral features, so thatmany parts of the arguments we shall develop must be by assertion: but then, that istrue of other, more conventional, approaches too. Electronic absorption spectra of a few typical transition-metal complexes areshown in Fig. 2-1. The following features are to be noted.

a. All absorptions are broad, often up to 2000 crrr1 wide yet occasionally down to 100 cm" 1 . Atomic line spectra are of the order 1 cnrr1 in width.b. Most bands in the near IR, visible and near UV are weak and about 102 to 104 times weaker than bands characterizing dyestuffs. These are called 'd-d' bands.c. Often, much more intense bands occur at higher energies, usually in the ultraviolet region. These comprise so-called 'charge-transfer' bands as well as ligand-centred η-τ(* and η-π* transitions.d. The spectra of most octahedral complexes of ions with the configurations d1, d4, d6 and d9 are characterized by a single absorption, while those for many cor- responding J2, J3, d1 and J8 complexes have up to three main absorptions.e. The spectra of d5 complexes show a large number of very weak absorptions, some of which are relatively sharp.f. The perceived colours of these complexes by transmission are those complementary to the absorptions. Suppose white light impinges upon a sample from a direction labelled z. At appropriate absorption frequencies, the electronic

Figure 2-1. Typical absorption spectra of transition-metal complexes.

2.2 The Valence Shell 23

arrangements in the molecules change as energy is absorbed. About 10 ~18 seconds

later, the same energy (frequency) is re-emitted and the ground state electronic arrangement is recovered. However, the light is emitted equally in all directions normal to the incident electric displacement. When viewed along direction z, less of the absorbed frequencies are observed than if no resonance had occurred and we record a net absorption in our spectrum. Further, the colour that we observe with our eyes is, of course, determined by absorptions occurring only in the visible part of the electromagnetic spectrum. With the latter point in mind, we note the colours of the permanganate ion (deep purple) and of the tetrachloro- and tetrabromocuprate(n) ions in Cs2[CuCl4] and Cs2[CuBr4] (yellow and brown). That for the tetrabromo complex is rather intense because the origin of the charge-transfer band lies lower in energy than that for the tetrachloro complex and we could describe the brown colour of Cs2[CuBr4] as a sort of 'red-black'.

Box 2-1 Spectrometers frequently record spectra on a wavelength scale (nm). This is because dispersion by gratings and prisms is more nearly linear in wavelength than in frequency. On the other hand, frequencies of transitions are directly related to the energy changes which are of more chemical significance. We shall report transition energies throughout this book on frequency scales. Frequency and wavelength are reciprocally related and 10,000 wavenumbers (cm"1) = 1000 nm.

There are many more details to be recognized within even the spectra illustratedin Fig. 2-1: sometimes, we observe bands which have split into two or morecomponents, so that some of the generalizations above are spoilt. We shall look intothese matters in some detail in due course. For the moment, there are two mainfeatures of all 'd-d' spectra upon which we must focus:

1) 'd-d' bands are relatively weak, and

2) the number and patterns of 'd-d' absorptions are characterized by the molecular geometry and by the dn configuration.

We shall return to 1) in Chapter 4. Here we consider the significance of the dn

configuration.

2.2 The Valence Shell

The species discussed so far belong to the class we might label Werner-typecomplexes. We use this description to differentiate from carbonyl-type or other lowoxidation state complexes. We stay with Werner-type complexes exclusively untilChapter 6. The radial waveforms for 3d, 4s and 4p orbitals of the metals in such24 2 Focus on the dn Configuration

complexes are shown qualitatively in Fig. 2-2 and emphasize a most importantpoint. This is that the 3d orbitals in Werner-type complexes are much more 'inner'than either the 4s or 4p orbitals. Though hardly core-like, the radial extension ofthe 3d orbitals is not great. Overlap of the metal 3d orbitals with ligand functions iscorrespondingly small. Before claiming that the extent of admixture of the d orbitalsinto the bonding molecular orbitals of a complex is also small, however, we mustconsider the relative energies of all orbitals involved. We expect the orbital energyordering for metals in higher oxidation states to be: ligand donor function < metal3d < metal 4s < metal 4p. On these grounds alone, metal orbital participation in anybonding molecular orbitals formed would be expected to decrease in the order 3d>4s> 4p as the energy separation between metal and ligand orbitals increases. But aswe have noted from Fig. 2-2, the ordering on overlap grounds would be 4p > 4s >3d. These trends are sketched in Fig. 2-3. We argue that while the energy matchingfavours strongest participation of the 3d function amongst the metal functions, theseorbitals are sufficiently withdrawn or contracted that their poor overlap with ligandfunctions leaves the metal 4s orbital as the dominant metal contribution in thebonding. This idea is a central thesis in this book. We shall re-emphasize the point againand again, and justify our position increasingly as we progress. Here, we make justone or two remarks about it. Firstly, we are not saying that the metal d orbitalsaren't involved in the orbitals that bind a complex, but merely that their participationis small. It is perfectly possible, however, to develop a consistent picture of chemicalbonding, spectral and magnetic properties together using, as a starting point, theidea that the d orbitals have effectively no participation in the bonding orbitals.This will seem a strange idea to many since the implication of all teaching texts, sofar as we are aware, is that "transition-metal chemistry is about the consequences of 2.2 The Valence Shell 25

d-orbital overlap." We consider that such a view sets up a false prejudice in themind of the reader and has engendered serious misunderstanding of the subject wecall 'ligand-field theory', as we shall discuss. It is to be acknowledged that ourassertion that the d-orbital participation in the bonding orbitals of a complex issmall leaves open the question of 'how small is small?'. As we shall see, however,even the limiting assumption of negligible participation of the d orbitals provides amost valuable viewpoint. So, with the promise to return to this seminal questionand to refine our position, let us now see something of what follows from theproposition. The proposition is that the bonds holding a Werner-type complex together aredominated by overlap of unspecified ligand orbitals with the transition-metal 4sorbital. The latter is, of course, spherically symmetric, so that the attractive(bonding) forces are largely undirected. This bequeaths to secondary repulsiveforces, like ligand-ligand repulsions, the determination of the molecular angulargeometry. Straightaway, therefore, the reason for the phenomenal success of Kepert'smodel, as described in the first chapter, is apparent. To be utterly simplistic aboutit: at this level, the metal doesn't care about the angular geometry, but the ligandsdo. The picture is very rough, of course, and still fails to explain the existence ofplanar complexes. We return to that question in Chapter 7. Two other, closely related, consequences flow from our central proposition. Ifthe d orbitals are little mixed into the bonding orbitals, then, by the same token, thebond orbitals are little mixed into the d. The d electrons are to be seen as beinghoused in an essentially discrete - we say 'uncoupled' - subset of d orbitals. Weshall see in Chapter 4 how this correlates directly with the weakness of the spectral'd-d' bands. It also follows that, regardless of coordination number or geometry,the separation of the d electrons implies that the dn configuration is a significantproperty of Werner-type complexes. Contrast this emphasis on the dn configurationin transition-metal chemistry to the usual position adopted in, say, carbon chemistrywhere sp, sp2 and sp3 hybrids form more useful bases. Put another way, while the 2s26 2 Focus on the dn Configuration

and 2p subshells together comprise the valence shell in carbon chemistry, the dsubshell of Werner-type complexes retains a free-ion-like integrity alongside a metalvalence shell of 4s (with some 4p) character.

2.3 The Roles of d Electrons

Surely a natural question to ask at this stage is 'if the d orbitals essentially don'toverlap with the ligand orbitals, what role, if any, do they play?'. Although there isan implication in that question that any role is minor, that is not the case at all. Thed electrons interact with the bonding electrons. Let us emphasize the word 'interact':it refers to a mutual action. The d electrons are affected by the bonding electronsand the bonding electrons are affected by the d electrons. We can progress a longway by considering these two aspects separately. Ultimately, to be sure, we mustrefine our arguments to make due recognition of the interaction. The effects of the bonding electrons upon the d electrons is addressed within thesubjects we call crystal-field theory (CFT) or ligand-field theory (LFT). They areconcerned with the J-electron properties that we observe in spectral and magneticmeasurements. This subject will keep us busy for some while. We shall return tothe effects of the d electrons on bonding much later, in Chapter 7.

Suggestions for further reading

1. The Roles of J-Electrons in Transition Metal Chemistry: A New Emphasis, M .

3.1 The Crystal-Field Premise

During the first twenty years or so of this century, an incredibly detailed

understanding of atomic line spectra was built up with the application of the, thennew, quantum theory. Indeed, the development of quantum theory came about inpart by the need to understand these spectral properties. We shall have to reviewsome basic features of the theory of atomic spectra for our present purposes, but weshall leave it for the moment. In the later 1920's, physicists, rightly flushed with their successes withinterpreting the rich, sharp spectra of atoms and gas phase ions, sought to extendtheir reach to the broader (and fewer) absorption bands that characterize the spectraof ions in crystalline matrices.* These bands occur at utterly different frequenciesto those of the corresponding free ions so that there is no similarity at all betweenthe spectra of free ions and of those in ionic or covalent lattices. Crystal-field theory (CFT) was constructed as the first theoretical model toaccount for these spectral differences. Tts central idea is simple in the extreme. Infree atoms and ions, all electrons, but for our interests particularly the Outer' ornon-core electrons, are subject to three main energetic constraints: a) they possesskinetic energy, b) they are attracted to the nucleus and c) they repel one another.(We shall put that a little more exactly, and symbolically, later). Within theenvironment of other ions, as for example within the lattice of a crystal, thoseelectrons are expected to be subject also to one further constraint. Namely, theywill be affected by the non-spherical electric field established by the surroundingions. That electric field was called the 'crystalline field', but we now simply call itthe 'crystal field'. Since we are almost exclusively concerned with the spectral andother properties of positively charged transition-metal ions surrounded by anions ofthe lattice,** the effect of the crystal field is to repel the electrons. Those electrons must not only avoid each other but also the negatively chargedanionic environment. In its simplest form, the crystal field is viewed as composedof an array of negative point charges. This simplification is not essential butperfectly adequate for our introduction. We comment upon it later.

* It is interesting that the very broad, so-called 'spin-allowed' transitions, like most of those inFig. 2-1, were not actually recognized as such until the 1950's. This was because of thecharacteristics of the spectrograph rather than the spectrometer.** To be contrasted with a negatively charged metal surrounded by positively charged groups.The idea of neutral ligands with donor lone pairs will be considered in due course.

Transition Metal Chemistry. M. Gerloch, E. C. Constable

3.2 Splitting of d Orbitals in Octahedral Symmetry

We are concerned with what happens to the (spectral) d electrons of a transition-

metal ion surrounded by a group of ligands which, in the crystal-field model, maybe represented by point negative charges. The results depend upon the number andspatial arrangements of these charges. For the moment, and because of the verycommon occurrence of octahedral coordination, we focus exclusively upon anoctahedral array of point charges.

Figure 3-1. The angular forms of the five d orbitals.

The set of five d orbitals share a common radial part like that sketched in Fig.2-2. Their angular parts are shown in Fig. 3-1. Let us consider the six point chargesin an octahedral array to be disposed along the positive and negative x, y and ζ axesto which these d orbitals are referred. This is conveniently drawn, as shown in Fig.3-2, by placing the charges at the centres of each face of a cube, itself centred onthe metal atom. By comparing the orbitals in Fig. 3-1 with the crystal field of pointcharges in Fig. 3-2, we observe that some orbitals are more directed towards thepoint charges than others. The dz2 and dx2_y2 orbitals are directed exactly towardsthe six charges while the dxy, dxz and dyz have lobes which lie between the ;c, y and ζ 3.2 Splitting of d Orbitals in Octahedral Symmetry 29

axes on which the charges are situated. Therefore, an electron placed in the dxyorbital will be less repelled by the crystal field than one placed in the dx2_y2 . It isobvious that the dispositions of the dxy, dyz and dxz orbitals with respect to the pointcharges are energetically equivalent. It is not obvious, but nonetheless true, that therepulsion suffered by an electron in the dx2_y2 orbital is the same as that by one inthe dz2 orbital (see Box 3-1). Altogether then, the energies of the five d orbitals (strictly of the electrons withinthem) in octahedral symmetry separate into two groups as shown in Fig. 3-3. All dorbitals are raised in energy by repulsion in the crystal field, but two go to higherenergies than the other three. Since spectroscopy and, indeed, most other d electronproperties of interest to us are concerned with relative energies, or splittings, ratherthan with absolute energies, a more usual representation of the differential crystal-field effect upon d orbitals is that shown in Fig. 3-4. We draw the energy levelswith respect to the mean energy of the whole d orbital set. This so-called

d xy> dxz >dyz

Figure 3-3. Two d orbitals are raised in energy more than the other three.30 3 Crystal-Field Splittings

Box 3-1 Each lobe of the dx2_yi orbital interacts predominantly with one point charge. The repulsive effects relate to the electron density within any given orbital so we might describe the interaction in units of 'lobe repulsion' and say that, for the dx2_y2 orbital, this amounts to 42 = 16 repulsion units (4 squared because electron density °c ψ2). The dz2 orbital can be written as a linear combination of two different orbitals which (Eq. 3.1) look like the dx2_y2 orbital but referred to the xz and xy planes: dz2_x2 and dz2_y2

= (dz2_x2 (3.1)

This identity is sketched below.

The dz2_x2 and dz2_y2 orbitals each interact with four point charges in precisely the same way as does the dx2_y2 orbital. Again the repulsion relates to electron density, so the total interaction of the combination is (4/^)2 + (4/^)2 = 16 of our repulsion units. In other words, the dz2 and dx2_y2 orbitals are degenerate in octahedral symmetry.

d72

dz2_y2

'barycentre' rule means that if the splitting between the two subsets of orbitals islabelled Aoci9 the higher pair lie at an energy +0.6 Aoci and the lower trio at -0.4 Aoct.('Barycentre' means a 'centre-of-gravity' type of rule.) An older alternative labelfor the octahedral-field splitting is 10Dg where, in the literal crystal-field model wehave introduced thus far, q is the charge on each ligand and D is a quantity relatedto the geometry. We shall make no use of these old meanings and just refer to Dq asa sort of 'dipthong of consonants'. In this notation, the pair of orbitals lies at +6 Dqand the lower trio at -4 Dq. Both Aoct and 10Dg are in common use and we shallswap between them at will. The subsets of d orbitals in Fig. 3-4 may also be labelled according to theirsymmetry properties. The dz2/dx2_yi pair are labelled eg and the dxy/dxz/dyz trio as t2g.These are group-theoretical symbols describing how these functions transform undervarious symmetry operations. For our purposes, it is sufficient merely to recognizethat the letters a or b describe orbitally (i.e. spatially) singly degenerate species, erefers to an orbital doublet and t to an orbital triplet. Lower case letters are used forone-electron wavefunctions (i.e. orbitals). The g subscript refers to the behaviour of 3.2 Splitting of d Orbitals in Octahedral Symmetry 31

Box 3-2 Another way to view the barycentre rule is to consider first the bringing up to the metal of a spherical shell of negative charge which increases the energies of all five d orbitals equally. Then, in this notional picture, if the spherical shell of charge redistributes towards the apices of an octahedron, those orbitals directed towards those apices suffer a further repulsion and energy increase, while those directed in between, acquire a relative stability.

: + 0.6 4>ctor + 6Dq

these functions under inversion through a centre - gerade or even. As all d orbitalsare centrosymmetric, all d subsets are here labelled g. The subscript 2 in t2g givesfurther symmetry information. We do not require this here and must just accept thelabel as a name. For a transition-metal ion in an octahedral environment with a single d electron(configuration d1), the ground state arises when that electron is placed in the lowerenergy t2g subset. Upon absorption of an appropriate energy - Z\oct, the electron ispromoted into the higher energy eg subset. Redistribution of the electron within thet2g set involves no energy change and will take place spontaneously and continuouslybecause of the equivalence of the three cartesian directions in the octahedron. Thus,only one energy change within the d orbitals is possible and corresponds to thetransition t2g—> eg. Illuminating such d1 ions with light of varying frequency, as inspectroscopy, may bring about that single transition when the frequency hv is suchthat hv = A0C1. The 'd-d' spectrum of the d1 ions comprises this single opticaltransition. Yet another representation of the d orbital splitting in Fig. 3-4 is that shown inFig. 3-5. Here, we imagine that the charges on the six ligands are smoothly variedor that their distances from the metal atom are so varied or that the radial extensionof the d orbitals is changed; or, of course, any combination of these. The splittingpattern (t2g + eg) remains unchanged for it is a property of the octahedral dispositionof the point changes. The magnitude of the splitting parameter Z\oct, however changesas the strength of the repulsions between d electrons and point charges changes. Assuch A0Ct thus measures the strength of the crystal field. Now suppose the system of metal and charges we have discussed represents ametal ion complexed by six ligands. These vibrate continuously. One such vibration32 3 Crystal-Field Splittings

slope = +6

slope = -A

limits during vibration

Figure 3-5. The variation of the octahedral splitting with respect to smoothly changing themagnitude of Aoct.

involves the bonds lengthening and shortening together - a so-called 'breathing'

mode. When the ligands are closer to the metal, the repulsions suffered by the delectrons are larger than when the ligands are more distant. So, during the course ofthe vibration, the crystal field strength varies between the limits indicated in Fig.3-5. Such vibrations typically take place in about 10~ 13 sec. An optical eventinvolving absorption and reemission between the t2g and eg subsets takes place inabout 10~18 sec. A beam of light, incident on a sample, therefore effectively 'sees'stationary molecules (the Frank-Condon principle). In a sample containing manysuch molecules, however, the light encounters molecules in every part of theirvibrational cycles and so we observe electronic 'd-d' transitions at frequencieseverywhere between the extreme vibration limits indicated in Fig. 3-5. Simple theorypredicts that Aoct is inversely proportional to the fifth power of the bond length(more sophisticated calculations actually give rather similar results). The energyspread indicated in Fig. 3-5 is therefore actually rather large and 'd-d' transitionsare typically observed as broad bands, maybe 2000 to 3000 cm"1 wide.

3.3 Splitting of d Orbitals in Tetrahedral and Other

Symmetries

Tetrahedral SymmetryPerhaps only slightly less common than octahedral symmetry is tetrahedralsymmetry. We now examine the d orbital splitting in this environment. The story ismuch the same as above, except that it is now convenient to place the four pointcharges of the tetrahedron as shown in Fig. 3-6. Here ligands are put at alternate 3.3 Splitting of d Orbitals in Tetrahedral and Other Symmetries 33

Figure 3-6. The angle subtended by a lobe of dx2_y2 and M - L is greater than that subtended bya lobe of dxv and M-L.

corners of a cube centred about the metal. We retain the same axis frame so that jc,y and ζ are again directed towards the midpoints of the cube faces. The dxy anddx2_y2 orbitals are included in the figure for discussion (the shading has nosignificance other than to differentiate these two orbitals). We note that the lobes of dxy are directed towards the midpoints of the cubeedges, whereas those of dx2_y2 point towards the midpoints of the cube faces. It isapparent that the lobes of dxy are oriented more nearly towards the point chargesthan are those of dx2.y2. An electron in dxy is thus repelled more than one in dx2_y2.Once again, it is obvious that the situations for dxz and dyz electrons are entirely likethat for dxy (permute the axis labels again) so these orbitals form a subset of three.Less obviously, the φ and dx2_y2 are equivalent and form a subset of two. Thesplitting diagram for the five d orbitals in a tetrahedral crystal field is shown in Fig.3-7. Once more, orbital energies are indicated with respect to a barycentre rule. Thesplitting is called ^tet (or 10Dg - which can be confusing; see Section 3.11) and thetrio lies higher in energy than the pair. The d orbital subsets are labelled t2 and e.

Atet or WDq

dz2, dx2_y2\ e : - 0.6 Atet or - 6Dq

Note the absence of the g subscripts here. Although the d orbitals are still centro-symmetric, the tetrahedral environment lacks a centre of inversion. The d orbitalsare therefore not classified with respect to a symmetry element which doesn't exist:the absence of the g subscript does not imply the opposite - i.e. u (ungerade orodd). Overall, then, we observe that the orbitals are inversely split in the tetrahedronwith respect to the octahedron. However, the differentiation between the subsets inthe octahedron was based on orbitals being oriented directly at or between the pointcharges; in the tetrahedron, all d orbitals point between the ligands, though someare closer to the point charges than others. Accordingly, the magnitude of thetetrahedral splitting is less than that in the octahedron. Simple geometricalcalculations show that, for the same metal (same d orbital radial functions) and forthe same bond lengths, these splittings are related by the expression in Eq. (3.2).*

Act - 4/9 ZLt (3.2)

In practice, these conditions of radial waveforms and bond lengths will not bemet exactly, so that a rough rule is that Atet ~ 0.5 z\oct in real systems. Once again,only one electronic, 'd-d\ absorption is expected (and observed), although muchshifted towards the red relative to that in an analogous octahedral complex.

Other EnvironmentsThe splitting patterns in crystal fields of symmetries other than octahedral ortetrahedral can be worked out using broadly similar principles. In general, the d-orbital degeneracy is raised (meaning decreased!) even more and there result up tofive energetically discrete subsets. That in turn begets more splitting parameterslike z\oct and often the situation ceases to be simple (see Box 3-3). We shall look atsome of these situations briefly in later chapters. For the moment, it suffices torestrict our concern to the so-called 'cubic symmetries' - octahedral and tetrahedral.So many complexes possess these symmetries, or at least approximately, that thisrestriction is not too serious at this stage.

3.4 Holes: d1 and

The discussion above might have pertained for example, to the energies andelectronic spectra of titanium(m) compounds. The same ideas can be applied withjust one modification to the J-electron properties of copper(n) complexes and other

Some prefer to write Atet = - % Aocl in order to emphasize the inversion of t and e orbitalsubsets. However, if Aoct and Zitet are defined as the orbital splittings, it is probably best to omitthe sign. 3.4 Holes: d1 and d9 35

Box 3-3 As an example of the effects of lower symmetry, consider the splitting of the d-orbital energies in a tetragonally elongated octahedron. This could arise either through the obvious arrangement of two long trans contacts (bond lengths) and four short, or by ligand dissimilarities as in a trans MA4B2 complex.

The weaker field along the B...B direction

repels a dz2 electron less than that in the A4 plane, and destabilizes the dxz/dyz pair less than the dyv:

free octahedral tetragonally

configuration t^e\ as shown on the left side of Fig. 3-8. This electronic arrangement,or configuration, clearly corresponds to the electronic ground state for the lower-lying t2g orbital set being filled while the higher-lying eg set is incompletely filled. On absorption of an energy A>ct, one of the t2g electrons will be promoted into theeg set, as on the right side of Fig. 3-8. As the eg set is now full, no further electronicpromotions are possible so that this corresponds to the one and only excited state ofthe octahedral d9 configuration. We thus observe a single absorption band in the'd-d' spectrum. The excitation t2ge| —> t2geg is equivalent to the transfer of the holein the eg3 configuration into the t2g set. We may view a transition in a d9 complex asa redistribution of a single hole (lack of electron) within a full d shell. However,while the transition of a single electron in octahedral d1 complexes involves theshift t2g —> eg, the transition of a single hole in the corresponding d9 system involvesthe shift eg —> t2g. A similar inversion follows for tetrahedral complexes. We shall36 3 Crystal-Field Splittings

hv=Aocl

Figure 3-8. The electronic transition for octahedral d9 ions.

have more to say about this hole formalism later. For the moment, it suffices tonote that octahedral and tetrahedral complexes of dl or d9 configurations give riseto just one electronic transition in their 'd-d' spectra. Some of the general featuresof Fig. 2-1 are thus explained. Unfortunately, the explanation of the greater numberof transitions in, say, d2 complexes is more difficult and lengthy. We must look atthat, however: it is our next topic. In mastering it we shall enter quite deeply intothe subject of crystal-field theory and learn more about the energetics of d electronsin transition metal complexes.

3.5 More Transitions for d2

Let us look now at the case of a d2 ion in octahedral symmetry. The orbital splittingis again as given in Fig. 3-4. With two d electrons, however, rather more electronicarrangements within the t2g and eg subsets are possible. At first sight there are three:both electrons may be housed in the t2g subset, both in the eg or one electron ineach, as indicated on the left of Fig. 3-9. In fact there are many more and they arisefor two reasons. The first concerns spin, since the electrons could either share acommon spin or their spins could be opposed. In the former case, the total spinquantum number for the pair of electrons is (!/2 + Vz) or 1 while, in the latter, it is( l / 2 - {/2) or O. The two-electron states associated with these spin quantum numbersare called (spin) triplets or singlets respectively. For those somewhat unfamiliarwith these labels, we provide a brief review in the next section. Now, in this section,we consider only those electronic arrangements of maximum spin. We do this fortwo reasons. Firstly, Hund's first rule defines the ground term of a free ion to beone of maximum spin multiplicity (see Section 3.6). Secondly, as will be discussedin Chapter 4, electronic transitions between states of the same spin multiplicity aremuch more allowed (the spectral bands more intense) than those involving a changeof spin. So we focus here on the spin-triplet states. However, even discarding the 3.5 More Transitions for d2 37

24«

strong-field after inclusion electronic

configurations of interelectron transitions from repulsion the ground term

Figure 3-9. Four spin-triplet terms arise for d2.

spin-singlet states, there arise four discrete types of spin-triplet arrangement ratherthan the three one might at first anticipate. Let us see why. Firstly, consider the spatial degeneracies of spin-parallel electronic arrangementswithin the configurations t|g, tlgeg, e\> Parallel spins must be placed in differentorbitals, of course, because of the Pauli exclusion principle. So spin-parallelarrangements within the t|g configuration necessarily involve one up-spin (say)electron in each of two of the members of the t2g orbital set. There are three sucharrangements. (The same result is to be had by noting that the empty orbital can beany one of three). We label this group of arrangements by the crystal-field termsymbol, 3 rj g (see Box 3-4). Next we consider the configuration e\. A spin-parallel arrangement must involveone electron in each of the two members of the eg subset. Ignoring spin (for we

Box 3-4 The left superscript indicates that the arrangements are all spin triplets. The letter Prefers to the three-fold degeneracy just discussed and it is in upper case because the symbol pertains to a many-electron (here two) wavefunction (we use lower-case letters for one- electron wavefunctions or orbitals, remember). The subscript g means the wavefunctions are even under inversion through the centre of symmetry possessed by the octahedron (since each d orbital is of g symmetry, so also is any product of them), and the right subscript 1 describes other symmetry properties we need not discuss here. More will be said about such term symbols in the next two sections.38 3 Crystal-Field Splittings

Box 3-5 Again the left superscript indicates the spin-triplet nature of the arrangement. The letter A means that it is spatially (orbitally) one-fold degenerate and it is upper-case because we describe two-electron wavefunctions. The subscript is g because the product of d orbitals is even under the octahedral centre of inversion, and the right subscript 2 must remain a mystery for us once again.

have fixed that as spin-parallel), there is only one way of filling this orbital subset.This unique arrangement is labelled by the term symbol 3 A 2g (see Box 3-5). As the energy of the eg orbital set in the octahedron is larger than that of the t2g,the energy of the configuration e\ is greater (by 2z\oct, that is, Zioct for each electron)than that of t|g. It should not be surprising that the energy of the 3 A 2g term (group ofwavefunctions) is higher than that of the 3Γ1§ term discussed above. Note, however,that the energy of the 3 A 2g term is not determined solely by the 2Aoct promotion aswe shall see. We have left the configuration t2lgeg till last because it involves some new ideas.Actually, as will be apparent, all three configurations involve exactly the sameprinciples. While their variety is not immediately apparent from our discussion ofthe el and t2g configurations, it is with the t2lgelg. Now the t2lgeg configuration symbolmeans that one electron is to be placed within the t2g subset and one within the eg.We have agreed to consider only those arrangements with parallel spins. The t2gelectron may be housed in any one of three orbitals while independently the egelectron may occupy one of two. Altogether therefore, there are (3 χ 2) = 6 spatialarrangements for these two electrons. However, the six arrangements are notdegenerate. They form up into two sets of three, with the term symbols 3T2g + 3T\g(the rules of the labelling are now hopefully clear enough). Why are the energies ofthese two terms different? Also of note is that the difference in their energies is notat all trivial, being around twice the magnitude of the crystal-field splitting zloct inmany systems. The answer has been given in Section 3.1. We have focused upon crystal-fieldenergies, that is, upon the need of the metal d electrons to avoid the regions ofhigher negative charge in the crystal (or molecular) environment. With more thanone d electron (or hole), as here, we must not forget that these electrons also needto avoid each other. Our discussion in the present section thus far has omittedconsideration of interelectron repulsion energy. Recall our caveat when asserting 3 3that the relative energy of the Γ1§ (from t2g) and the A 2g (from e%) is not givensimply by Aoci, the energy splitting of the t2g and eg orbitals. That is because theelectron - electron repulsion energies for these two arrangements are not the same.They are not the same because the spatial proximity of members of the eg orbitalpair is not the same as that for members of the t2g set. Similarly, the proximities andinterelectron repulsion energies vary within the electronic arrangements of the t2gegconfiguration, as we now discuss. Taking one electron from each of the t2g and eg subsets, we can form high- andlow-energy spatial triplets, 3T2g and 3 r lg , as shown in Eq. (3.3) (we write xy for dxy,etc). 3.5 More Transitions for d2 39

(3.3)

(xy)(x2-y2) (y (xz)(z2)

The 3r2g term wavefunctions lie lower in energy than those of the 3 r l g .Qualitatively, we can rationalize this energy ordering most easily by consideringthe first component of each of these terms. The relevant pairs of orbitals are shownin Fig. 3-10. The orbital pair (xy)(x2-y2) is obviously much more crowded togetherthan is the (xy)(z2) pair and so the interelectron repulsion energy associated with theformer arrangement is much greater than with the latter.

On the right side of Fig. 3-9 are represented the relative energies of the two 3 T l gterms, the 3T2g and 3 A 2g . The ground term is the 3Γ1§ from the t2g configuration. Spin-allowed electronic transitions (those between terms of the same spin angularmomentum - but see also Sections 3.6, 3.7 and Chapter 4) now take place uponexcitation from 3rlg-^3r2g, ->3A2g, ->3Γ1§. The 'd-d' spectra of octahedrallycoordinated d2 ions thus exhibit three bands. Similar arguments for tetrahedrallycoordinated d2 ions yield three transitions also, but this time from a 3A2 groundterm: 3A2->3Γ2, -^r1, —>37V Clearly, similar results apply to those J8 ions havingtwo holes in a full d shell rather than two electrons. We shall look at these holeequivalencies more carefully in Section 3.7. The approach we have adopted for the d2 configuration began from the so-called 'strong-field' limit. This is to be contrasted to the 'weak-field' scheme that wedescribe in Section 3.7. In the strong-field approach, we consider the crystal-fieldsplitting of the d orbitals first, and then recognize the effects of interelectronrepulsion. The opposite order is adopted in the weak-field scheme. Before studyingthis alternative approach, however, we must review a little of the theory of free-ionspectroscopy40 3 Crystal-Field Splittings

3.6 Atomic Orbitals and Terms

The present section is offered as a review of the jargon of the theory of free-ionspectroscopy with little in the way of any free-standing explanation.* 1 A transition metal with the configuration J is an example of a 'hydrogen-like'atom in that we consider the behaviour of a single (d) electron outside of anyclosed shells. This electron possesses kinetic energy and is attracted to the shieldednucleus. The appropriate energy operator (Hamiltonian) for this is shown in Eq.(3.4).

= - ^ H-like ~ ~V — (3.4) 2m r

Solutions to the Schrodinger equation (3.5) are called one-electron wavefunctions or

orbitals and take the form in Eq. (3.6)

^Η-ΜβΦ ~ £ Φ ^'^

θ ω} (3-6)

The radial functions, R depend only upon the distance, r, of the electron from thenucleus while the angular functions, γ1 ($ 9 <τ)), called spherical harmonics, dependonly upon the polar coordinates, θ and φ. Examples of these purely angularfunctions are shown in Fig. 3-11.

1V V V V OO -* 10 ·* 20 ·* 30

Figure 3-11. The shapes of the angular functions are determined only by the theory of angularmomentum in spherical symmetry.

The orbitals are labelled Is; 2s, 2p\ 3s, 3/7, 3d; etc. to indicate the principalquantum number η (here equal to 1, 2, 3 ) and the orbital angular momentumquantum number, /, according to the code in (3.7).

This topic is described fully, but at the same level as adopted in the present book, inOrbitals, Terms and States'. 3.6 Atomic Orbitals and Terms 41

1 2 3 4 5 ... (3.7) P d f g h

Associated with each / value are (21 + 1) values of m/, ranging /, /-1 -/,describing the ζ component of the angular momentum. Thus, we find one s function,three ρ functions, five d functions, and so on. The Hamiltonian (3.4) is a function of the usual spatial coordinates (;c, _y, ζ or r,0, 0). Electrons possess the intrinsic property of spin, however, which is to bethought of as a property in an independent, or orthogonal, space (spin space). Spinis actually a consequence of the theory of relativity but we shall merely graft on theproperty in an ad hoc fashion. The spin, s, of an electron (don't confuse with sorbitals!) takes the value 1/2 only. The ζ component of spin, ras, takes (2s + 1)values of ras, ranging s, s-l,...-s. Thus for the single electron, ras = +1/2 or -1/2,also labelled a or j8, or indicated by T or i.Now consider a d2 ion as an example of a so-called 'many-electron' atom. Here,each electron possesses kinetic energy, is attracted to the (shielded) nucleus and isrepelled by the other electron. We write the Hamiltonian operator for this as follows:

il (3.8)

where each pair of the η electrons in the dn configuration suffer mutual repulsionsthat are inversely proportional to the instantaneous distance, r,·,·, between them. Thesum in the second part is for / < j in order not to count these pair-wise interactionstwice and to prevent the ith electron repelling itself. This second operator is calledthe Coulomb operator. Solutions, Ψ for this Hamiltonian,

X^V =W (3-9)

are called 'many-electron' wavefunctions, or term wavefunctions, because they

describe the behaviour of many (n) electrons as a group. They are not orbitals.These groups of wavefunctions - terms - possess the qualities of orbital- and spin-angular momentum, just like the orbitals of (3.5), however. Their orbital angularmomentum is labelled by L, according to the code:

L= O 1 2 3 4 5 ... (3.10) S P D F G H

and associated with each L value are (2L +1) values of ML, referring to the ζcomponents of the orbital angular momentum. The spin angular momentum of a term is labelled with S and may take integrally nseparated values based on O or 1/2 depending upon the d configuration; viz. 5 = 0,1, 2... or S = 1/2, 3/2, 5/2... Associated with each such S value are (2S + 1) valuesof MS for the ζ components of spin angular momentum, with M5 taking the values 5,S-I. ..-S.42 3 Crystal-Field Splittings

The total form of the many-electron wavefunctions, ψ, of (3.9) can be computed

for free ions but only after lengthy numerical procedures. Let us imagine this tohave been done and ask what angular momentum properties are assocated withthese solutions. We find them aggregating into groups - terms - characterized byappropriate pairs of orbital- and spin-angular momenta: L, S. Instead of so labellingthem, they are conventionally described by term symbols of the form 25+1L. Examplesare 2D, 3F, 6S etc, pronounced doublet D, triplet F, sextet S, etc. Each of these terms is(2L+ l)(2S +l)-fold degenerate because there are (2L +1) ML values for each L and(2S + 1) MS values for each S; and spin- and orbital-angular momenta are indepen-dent properties (in the absence of spin-orbit coupling). The degeneracies of 2D, 3F,6 S terms, for example, are 10, 21, 6. Note that, throughout this discussion, we have used lower-case letters whenrefering to orbitals and upper-case when we mean many-electron wavefunctions.There arises the question of, 'what are the relationships between / and L, or betweens and S T. They are determined by the vector coupling rule. This states that theangular momentum for a coupled (i.e. interacting) pair of electrons may take valuesranging from their sum to their difference (Eq. 3.11).

L ->/! + /2, /!+ /2-1, .... I / ! -Z 2 I (3.11)

The same* goes for spin angular momentum (Eq. 3.12).

S—>Si + S29 s\ + S2 - 1, . . . .Is] - S2I (3.12)

The rule means that the angular momentum of each of a pair of electrons may beparallel - meaning about the same physical axis and in the same sense - or opposed,and that the quantum condition allows only integrally separated values betweenthese limits. For the d2 case, the / value for each electron is 2 and so L can take thevalues 4, 3, 2, 1 or O, corresponding to the labels G, F, D, P, S. The s value for eachelectron is 1/2, so S can take values 1 or O and there may thus arise terms 3G, 3F,3 3 3 1 1 1 1 1 D, P, S', G, F, Z), P, S. However, not all these terms do arise for the 'equivalent' 2d electrons of the d configuration, because some electronic arrangements violatePauli's exclusion principle while others ignore the fact that electrons are 2indistinguishable. For the d configuration, the first electron may have any one offive mi values and any one of two ms - or 10 possibilities altogether. The secondelectron is left with only 9 choices of the (m\, ms) combination because of theexclusion principle - hence yielding 90 arrangements for the pair. However, theindistinguishability of electrons means we cannot assign meaning to the adjectives'first' and 'second' here and have thus counted each arrangement twice. Altogether,therefore, the d2 configuration is 45-fold degenerate. It can be shown to give rise tothe term set: 1G, 3F, 1D, 3P, 1S. The (2L + 1)(25 + l)-fold degeneracies of theseterms are 9, 21, 5, 9 and 1 respectively, adding to 45, as required.

* The vector coupling rule applies to all forms of angular momentum:

Figure 3-12. Free ion terms arising from the d2 configuration.

All this is summarized in Fig. 3-12. The energy ordering of the free-ion terms isnot determined by consideration of angular momentum properties alone and ingeneral yields only to detailed numerical computation. The ground term - and onlythe ground term - may be deduced, however, from some simple rules due to Hund.

Hund's first rule: The ground term will be one of maximum spin- multiplicity (maximum S)

Hund's second rule: If ambiguity remains, the maximum-spin ground term will then be one with maximum L

2 3 3 So for d , the ground term is a spin triplet, and F rather than P. Let us recapone or two points. A configuration describes an orbital assembly in which norecognition is made of the Coulomb interaction between electrons: there are 45equally good (equi-energetic) ways of arranging two d electrons with regard to theirkinetic energy and attraction to the nucleus. When we recognize that these electronsrepel and otherwise interact through the Coulomb operator, we observe the termsplitting on the right-side of Fig. 3-12. There are now 21 ways of equal minimumenergy in which two d electrons possess kinetic energy, are attracted to the nucleus,and avoid each other (as best they might). For the (assumed) ordering in Fig. 3-12,44 3 Crystal-Field Splittings

we note that there are 9 second-best ways of satisfying these three constraints, andso on. The 21 best ways happen, as it were, to share the common angular moment-um properties, designated by the term symbol 3F, and so on again. Each member(state) of each term is a two-electron wavefunction, describing one or anotherparticular arrangement of exactly two electrons. The electrons have coupled through the agency of electrostatic (repulsive)interactions. Electrons may also couple via the (independent) means of magneticinteraction we call spin-orbit coupling. This effect is generally smaller than theelectrostatic coupling and we shall largely ignore it in this book. Spin-orbit couplingis not unimportant, especially when one comes to consider the magnetic propertiesof transition-metal compounds. However, we return now to our main enquiryconcerning the effects of crystal fields upon these wavefunctions.

3.7 Crystal-Field Splitting of Free-Ion D Terms

Term wavefunctions describe the behaviour of several electrons in a free ion coupledtogether by the electrostatic Coulomb interactions. The angular parts of termwavefunctions are determined by the theory of angular momentum as are the angu-lar parts of one-electron wavefunctions. In particular, the angular distributions ofthe electron densities of many-electron wavefunctions are intimately related to thosefor orbitals with the same orbital angular momentum quantum number; that is,

Box 3-6 The electron distributions in term wavefunctions and orbitals may be the same or complementary, as shown below

(a) (b)

Y electron density possible forms of the electron in a density in the corresponding d orbital part of a D term

The identity of (b) with (a) is obvious. In (c), a section of the density is shown to take the form of a spherical density from which a density of the form (b) has been subtracted. Alternatively, (c) may be viewed as a distribution of positive charge in the form of (b). Whether components of a D term take the form (b) or (c) depends upon the number of electrons described by the many-electron wavefunction. 3.7 Crystal-Field Splitting of Free-Ion D Terms 45

when L = /. So the shapes of electron densities for the five members of a D termare closely related to the shapes of the densities in the five d orbitals. In an octahedral crystal field, for example, these electron densities acquire diffe-rent energies in exactly the same way as do those of the d-orbital densities. Wefind, therefore, that a free-ion D term splits into T2g and Eg terms in an octahedralenvironment. The symbols T2g and Eg have the same meanings as t2g and eg, discussedin Section 3.2, except that we use upper-case letters to indicate that, like theirparent free-ion D term, they are generally many-electron wavefunctions. Of coursewe must remember that a term is properly described by both orbital- and spin-quantum numbers. So we more properly conclude that a free-ion term 2S+1D splitsinto 25+1Eg + 2S+1 r 2g in octahedral symmetry. Notice that the crystal-field splitting hasno effect upon the spin-degeneracy. This is because the crystal field is definedcompletely by its ordinary (x, y, z) spatial functionality: the crystal field has no spinproperties. Consider, for a change, the ground term of the d4 configuration. A quick way ofdetermining the free-ion ground term (and only the ground term) is as follows.Himd's rules require that the ground term be of maximum spin multiplicity. Wedraw a set of five boxes to represent the five m/ values for d orbitals placing thefour electrons in parallel

mi= 2 I O -1-2

T T T T

spin formation. Pauli's exclusion principle then requires that these electrons occupydifferent d orbitals (different mi values). The total ζ component of orbital angularmomentum M L is just the sum of the individual m/ values, 2 + 1 + 0 - 1 = 2.All other possible arrangements of four parallel electrons within the set of fiveboxes yield ML values in this way which range from 2 to -2. Hence L for the set is2, i.e. D. The total spin quantum number M5 is X(ra,)j = 4 x 1 / 2 = 2 and hence S = 5 52, The ground term is therefore D. In an octahedral field the D term splits to give5 5 T2g + Eg octahedral-field terms. (They are still called 'terms' because they are allmany-electron wavefunctions). 1 Finally, consider also the case of the d configuration. The ground (and in this 2case, only) term, worked out as above, is Z). The reader may object that for d\ weshould not use upper-case labels because we are dealing with a single electronrather than with a many-electron wavefunction. But we can do so, because the word'many' subsumes the particular case of One'. To say that the 2D term of d1 splitsinto 2r2g + 2Eg in an octahedral crystal field is merely to put our knowledge of thed—> t2g + eg splitting onto a uniform basis so that we can compare all dnconfigurations. There are other advantages too, as we shall see in Section 3.10.46 3 Crystal-Field Splittings

3.8 Crystal-Field Splitting of Free-Ion F Terms

2 3 8 3 3The ground term of the d configuration is F. That of J is also F. Those of d and 1 4d are F. We shall discuss these patterns in Section 3.10. For the moment, we onlynote the common occurrence of F terms and ask how they split in an octahedralcrystal field. As for the case of the D term above, which splits like the d orbitalsbecause the angular parts of their electron distributions are related, an F term splitsup like a set of/orbital electron densities. A set of real / orbitals is shown in Fig.3-13. Note how they comprise three subsets. One set of three orbitals has majorlobes directed along the cartesian χ or 3; or ζ axes. Another set comprises threeorbitals, each formed by a pair of 'clover-leaf shapes, concentrated about two ofthe three cartesian planes. The third set comprises just one member, with lobesdirected equally to all eight corners of an inscribing cube. In the free ion, of course,all seven / orbitals are degenerate. In an octahedral crystal field, however, the

χ or y or ; x or ν or ζ

Figure 3-13. Angular forms of the seven/orbitals.

3.8 Crystal-Field Splitting of Free-Ion F Terms 47

electron densities associated with these orbitals are differentially repelled. Thosefor the first set are repelled most because the orbital lobes are pointing directly atthe various ligands. Those of the second set are equivalent amongst themselves andare repelled less since they point between the ligands. The unique/-orbital densityis the least repelled because each lobe is even further from the ligand sites. All thisis summarized in Fig. 3-14, showing the splitting of/orbitals in an octahedral fieldas Fig. 3-4 showed the equivalent splitting for d orbitals. The orbital subsets arelabelled f l u , t2u, a2u respectively for the three sets described above. All /orbitals areodd (ungerade) under inversion through the centre of symmetry possessed by theoctahedral field and so are labelled with the u subscript here.

flu

Figure 3-14. Splitting of/orbitals in octahedral symmetry.

F terms split analogously but, because we are here considering F terms arisingfrom dn configurations (we could, but don't, consider F terms arising from f"configurations, by the way), the many-electron wavefunctions are built from productsof d orbitals of g symmetry. Hence, the octahedral-field terms arising are necessarilyof g symmetry and so we get the result F^Tlg + T2g + A 2g . Another consequence ofthe difference between F terms built from J-orbital products versus/orbitals is thatthe sign or sense of the splitting depends upon the number of d electrons and mayor may not be the same as that shown for the / orbitals in Fig. 3-14. There arevarious ways of determining whether the 3F term from d2 splits with a 3Γ}§ term 3lowest in energy, or with the A 2g term lowest. Although we have not prepared thegroundwork in the present text to describe the more direct of these routes, we areable to decide the issue, however, by reference to the strong-field scheme. Thus, inthe case of J2, for example, we know from the discussion in Section 3.5 that thelowest energy arrangement in an octahedral field is orbitally three-fold degenerate.This establishes the term splitting for the d2 case as that shown in Fig. 3-15. Note,once again, that both free-ion and octahedral-field terms are all spin triplets.48 3 Crystal-Field Splittings

2 d

3 2Figure 3-15. Splitting of the F term arising from the d configuration in octahedral symmetry.

3.9 Free-Ion S and P Terms in Crystal Fields

An S term, like an s orbital, is non-degenerate. Therefore, while the effect of a

crystal field (of any symmetry) will be to shift its energy, there can be no questionof its splitting. The ground term for the d5 configuration is 6S. In an octahedralcrystal field, this is relabelled 6Alg; in tetrahedral symmetry, lacking a centre ofinversion, it is labelled 6 A 1 . The three ρ orbitals are directed along the three cartesian axes and so, in anoctahedral crystal field, suffer equal repulsion from point charges sited on thoseaxes. The energies of the three ρ orbitals, therefore, remain degenerate. Similarly, afree-ion P term remains unsplit in octahedral or tetrahedral crystal fields and islabelled Tlg or T1 respectively. The only spin-triplets arising from the d2 configuration are 3F (ground) and 3P.The effects of an octahedral or tetrahedral field upon these two terms aresummarized in Fig. 3-16. We note that three spin-allowed electronic transitions should be observed in the'd-d' spectrum in each case. We have, thus, arrived at the same point establishedin Section 3.5. This time, however, we have used the so-called 'weak-field'approach. Recall that the adjectives 'strong-field' and 'weak-field' refer to themagnitude of the crystal-field effect compared with the interelectron repulsionenergies represented by the Coulomb term in the crystal-field Hamiltonian,

H CF = -/** <~ CF (3.13)

where VCF is the so-called crystal-field potential. Its a question of the order in whichwe consider these two effects. In the weak-field scheme, we begin with the free-ionterms left after the Coulomb interactions and then consider a weak crystal-fieldpotential. In the strong-field approach, we begin with the strong-field configuration, 3.9 Free-Ion S and P Terms in Crystal Fields 49

d2

'2g

Figure 3-16. Octahedral field spin-triplet terms arising for d2.

recognizing the crystal-field splitting of the d orbitals, and then take account of theinteraction repulsion energies. Of course, the true state of affairs in any real metalcomplex lies somewhere inbetween the weak-field and strong-field limits. Thecontinuum between these limits is shown semi-quantitatively by the (partial)correlation diagram in Fig. 3-17. On the left are shown the effects of the progressi-ve introduction of the components of the crystal-field Hamiltonian, HCF in the ordergiven in Eq. (3.13) - the weak-field approach. On the right, these contributions aretaken in the order HH-iike, ^CF and Coulomb. The term energies inbetween the weak-and strong-field limits vary across the abscissa with respect to the relativemagnitudes of the Coulomb and crystal-field contributions. Unjoined lines in thediagram refer to spin-singlet terms which are otherwise omitted for simplicity. Thediagram indicates that, for d2 in octahedral symmetry, the ground 3Tlg term from 3Fcorrelates with the strong-field configuration t2§; that the 3T2g correlates with h\e\\3 A2g with e\\ and that the 3 r l g term arising from the 3P free-ion term - labelled3 Tlg(P) as opposed to 3Tlg(F) - correlates with t2lgelg. We have noted how the relative term energies vary across the diagram. Inparticular, while the energy separations between the three strong-field configurationson the right are zloct or 10Dg, the energy separations between the 3 r lg , 3T2g and 3A2gterms arising from 3F on the extreme left (weak-field limit) are 8Dq and WDq, asshown. So while the 3T2g and 3A2g terms stay 10Dg apart right across the diagram,the energy separation, 3Tig(F)<^>3T2g, varies from SDq to IQDq as the crystal field isincreased (note that this change is over and above that due to the variation in thevalue of Dq itself). This effect is often represented by the diagram in Fig. 3-18.On the left side of the diagram, the energy separation between the 3P and 3F termsof the free ion is denoted as 155, where 5 is a parametric measure of the magnitudeof the Coulomb interaction - in effect, 5 is for interelectron repulsion what Dq isfor the crystal field. It is not appropriate here for us to enquire further into thechoice of this, seemingly odd, symbolism.* Moving one step to the right in Fig. 3- 18, we see the crystal-field term energies in the weak-field limit. Note that theunsplit 3P term does not shift either. This is because VCp is defined with respect to abarycentre rule so that, as elsewhere in both Fig. 3-17 and 3-18, the crystal-fieldeffects ignore the overall shifts of Fig. 3-3 and refer just to splitting energies. Then,as we move to the right of Fig. 3-18, the energies of both 3Tlg terms shift by somesmall amount xDq. This is because these wavefunctions possess the same symmetryand can mix together (in the same way that two s type molecular orbitals in adiatomic molecule can mix together). As they mix, their energies change such thatthe energy of the higher one [37"lg(P)] increases while that of the lower one [3Tig(F)]decreases by the same amount. From Fig. 3-17, we know that the 3Tlg(F) takes an 3energy SDq less than T2g in the strong-field limit. Hence, χ in Fig. 3-17 lies betweenO and 2. Again, the actual magnitude of χ in any real system is a measure of therelative magnitudes of the crystal-field and interelectron repulsion effects. The

* Except to note that the occurrence of the coefficients 15 and 10 in \5B and IdDg obviate theneed for fractions here or elsewhere in crystal-field theory: thus, they are there for reasons ofconvenience and definition only. 3.9 Free-Ion S and P Terms in Crystal Fields 51

15B ~~ l·^

'2g

Figure 3-18. Interaction between

3 χ Dq rlg(F) and 3 r lg (P) terms.

3 TlgCP)

Figure 3-19. Schematic represen-

tation of the energy variations of the spin-triplets of d2.52 3 Crystal-Field Splittings

changes across Fig. 3-18 are not discontinuous, of course, but smooth and may be 3 3redrawn as in Fig. 3-19. The energies of the A 2g and J2g terms vary linearly with 3Dq with constant relative slopes of 10Dg. Those for the two T1g terms curve awayfrom one another (as if they repel each other). We shall refer to this diagram againin Section 3.11.

n3.10 Splitting Patterns for d Ground Terms

nA complete description of the effects of a crystal field upon a d ion would include nsimilar analysis of the behavours of all terms arising for that d configuration. Box 23-7 summarizes the case for J , and in Box 3-8, we illustrate a method of using Fig.3-19 to determine Dq and B values from real spectra. Much useful understanding of the processes of crystal-field theory, however, canbe had from a study of just the free-ion ground terms. Application of the simpleprocess in Section 3.7 identifies the ground terms for dn free ions as 2D, 3F, 4F, 5Z), S, 5D, 4F, 3F, 2D for η = 1 to 9 respectively. There are some interesting patterns in6

this series. Firstly, note that the ground terms for J10"" configurations are the sameas for dn. This is because a dl°~n configuration of electrons behaves in many wayslike a dn configuration of holes. There is no question of any inversion here, however,because two holes repel each other just as do two electrons. This symmetry betweenjio-n anc[ ^n configurations extends to all terms. It does not apply to the absoluteenergies of any of the free-ion terms, however, for the mutual shielding for 10 -nelectrons is not the same as for η electrons. Secondly, if spin is ignored, we alsoobserve a symmetry at the 1/4 and 3/4 periods*. Thus, the orbital (spatial) parts ofthe ground terms for d5+n are the same as for d'\ as also are those for d]0~n like d5~n.The reason for this can be seen in two ways. With the restriction to the ground term(i.e. of maximum spin multiplicity), d5~n involves η holes in the half-shell of five dorbitals. The number and manner of the arrangement of, say, three electrons amongstthe five d orbitals is the same as the number and manner of two holes: similarly for,say, d1 and d8 configurations. The other way to look at it, with the same restrictionof maximum spin multiplicity, is as follows: The configuration d5+n (maximum spin)is like that for dn plus a filled half-shell. Thus, for η = 1:

Tl T t t T is like I

Ignoring spin, the ground term for d5+n is the same as for dn. But, from the firstsymmetry, as described above, that J10"" is like dn, this implies that d5~n is like dnfor the ground term. By way of emphasis: the symmetry at the half-period - J10""

* The 1/4, 1/2, and 3/4 periods occur between d3 and d4; at d5; and between d7 and d8,respectively. 3.10 Splitting Patterns for dn Ground Terms 53

Box 3-7 Supplementing the knowledge we have so far about the splitting of S, P, D and F terms in octahedral fields is the fact that a G term gives rise to Alg + Eg + T2g + Tlg crystal-field components. The diagram in the figure below - a so-called 'Tanabe-Sugano' diagram - shows the energies of all octahedral-field terms arising from the d2 configuration as functions of Dq and B. By convention, energy levels are plotted with respect to the ground term. Curvatures in these variations result in part from mixing between terms of the same kind (same spin and spatial symmetry) and in part because this method of presentation has the ground term along the abscissa.

70

60

] S 3 50 T1

40 3 T,

30

1 G 3 F 1 D 10

3 T1 10 20 30 Dq IB

like dn - applies to all terms of these configurations while the symmetries at the1/4 and 3/4 periods apply only to the ground terms. The effects of an octahedral crystal field upon each of the ground terms is shownin Fig. 3-20. This diagram was constructed as follows. From our previousdiscussions, D^>T2g + E8, F^Tlg + T2g + A2g (with T2g always in the middle), andS—>A lg For the J1 configuration, 2T2g lies lower than 2Eg. The symmetries weobserved for the ground terms of the free-ion terms at the 1/4, 1/2 and 3/4 periodsalso apply for the crystal-field splittings except that, while two electrons (or holes)always repel one another for the Coulomb contribution, electrons are repelled bythe crystal field but holes are attracted. So these symmetries give the same resultsfor the two-electron Coulomb operator but inverse results for the one-electroncrystal-field operator. Applying these rules, the 2D of d9 yields 2T2s and 2E8 terms inthe opposite energy order to those of d1, i.e. 2Eg lower than 2T2g. The 5D term of d454 3 Crystal-Field Splittings

Box 3-8 The figure below abstracts just the spin-triplet part of the Tanabe-Sugano diagram in the previous box. Suppose we have recorded the electronic 'd-d' spectrum of [V(H2O)6J2+ and identified two out of the three possible 'spin-allowed' (triplet-triplet) bands at energies 17,200 cm- 1 and 25,600 cmr1

60

50

40

§ 30

20

10

3 Dq/B

The question arises 'to which of the transitions 3rlg->3r2g,—>3A2g,-»3'rlg(P) do these bands correspond?'. We proceed by determining at what value of DqIB on the abscissa is the ratio 256/172 = 1.49 reproduced by the plot: this is shown by the vertical broken line in the figure . Then we construct the horizontal broken lines to meet, as shown. For va we find EIB on the ordinate as 25.9, and for vb we get EIB = 38.7. From either we find, therefore, that B = 665 cm' 1 . On referring back to the vertical line we thus find IQDq = 18,600 cm -1 . At the same time we have established that va corresponds to the transition -^3r(2g)(F) and vb to —>3rlg(P). The transition ->3A2g(F) is predicted to lie at ca. 36,000 cirr1.

also splits inversely with respect to the 2D term of dl\ the 5D term of d6 splitsinversely with respect to d9 and hence the same way round as for the 2D of d1. Turning now to the F terms. We have previously established that the ground termof d2 in octahedral symmetry is 3Tlg. Therefore, by the symmetry in the 1/2 period,that for d8 is 3A2g and by the symmetries in the 1/4 and 3/4 periods, the groundterms for d3 and d1 are 4A2g and 47\g respectively. Finally, the 6S term of d5 does not split and is labelled 6Alg: d5 is its own hole-equivalent. While there seems to be a lot to remember in Fig. 3-20, all one needs tocommit to memory is the ordering for, say, d1 and d2. The hole formalisms - n 3.10 Splitting Patterns for d Ground Terms 55

1 /ι /ι

- S I I (N

I O U

PO

£ 3 W) Ξ56 3 Crystal-Field Splittings

inversions at 1/4, 1/2 and 3/4 periods - do the rest. Lastly, consider tetrahedralsymmetry. In Section 3.3, we saw how the d orbitals split with (dxz, dyz, dxy) higherin energy than (dzi, dxi_y2)\ in other words, inverted with respected to the octahe-dral- field splitting. One reason why some prefer to write (3.2) as Aia = -4/9zloct is toindicate that 'the tetrahedral crystal field is (4/9 times) the negative of the octahedralfield'. This remark is not intended to confuse but merely to provide background toour assertion now that all splitting diagrams in Fig. 3-20 are inverted for tetrahedralcrystal fields. Of course, the g subscripts are omitted in the tetrahedral case.

3.11 Orgel Diagrams

The information in Section 3.9 and Section 3.10, referring to the crystal-fieldsplittings of ground terms and all terms of the same spin multiplicity, can be veryneatly encapsulated within two famous diagrams due to Orgel. Somewhat analogous,

and more comprehensive, diagrams for the effects of crystal fields on all termsarising from a given dn configuration are called Tanabe-Sugano diagrams: anexample was given in a Box 3-7 above. The Orgel diagrams are shown in Fig. 3-21and 3.22. They refer to the octahedral- or tetrahedral-field splitting of ions with Dor F ground terms respectively. Ions with d\ J4, d6 or d9 configurations possessonly one term of maximum spin multiplicity - 2D or 5D, as appropriate. The crystal-field T2(g) and E(g} terms that arise are unique in each case and there are nointeractions between terms of the same symmetry as in Section 3.9. Thus, theenergies of these T2^ and E(g} terms vary linearly with WDq as shown in Fig. 3-21.The two sides of the figure depict the situations with either a T2(g) term or an E(g)term being lowest in energy. The slopes of the lines arise from the barycentre rulesso that the energy of the more degenerate T2(g) term slopes more gently than that ofthe E(g) term. This corresponds to the energies +6Dq and -4Dq in Fig. 3-4, forexample. The inversions in the 1/4, 1/2 and 3/4 periods as well as that on changingoctahedral for tetrahedral symmetry are all accounted for in the abscissa labels.Recall our remark in Section 3.3 that we sometimes refer to the orbital triplet-singlet splitting in both octahedral and tetrahedral symmetry as IQDq, rather than58 3 Crystal-Field Splittings

A)Ct or A^. The context should remove ambiguity. On the other hand, that veryambiguity means that Orgel's diagrams are always drawn with Dq varying alongthe abscissae, regardless of octahedral or tetrahedral symmetry. Equivalently, the gsubscripts are enclosed in parentheses here so that these diagrams serve bothgeometries. Similar rules of construction have been followed in Fig. 3-22. Here, the higherlying 3P or 4P terms (as appropriate) are included. Curvatures due to the repeated(mixing) T1(g) terms are shown, corresponding to the representation we havediscussed in Fig. 3-19. Note, however, that the interaction between terms of thesame symmetry is essentially inversely proportional to their energy separation: thecloser they are in energy the more they mix and 'repel' one another. Because of thevarious inversions we have described, the highest energy term arising from an Fterm can be the T1(g). In this case, the Tlg(F) and Tlg(P) would be brought into closeproximity and might be expected to cross with increasing crystal-field strength.Their interaction prevents their crossing (see also the 'non-crossing rule') and thecurvatures displayed in the Orgel diagrams in these circumstances are very great.

3.12 Concluding Remarks

In the next chapter we look at the intensities of 'd-d' electronic transitions. We

shall see that transitions between terms of the same spin-multiplicity are muchmore intense than those involving a change of spin. It is for this reason that ourfocus in the present chapter has been on the former. We have seen that for J1, J4, d6and d9 configurations in octahedral or tetrahedral environments, there is only oneso-called 'spin-allowed' transition. For d2, J3, d1 and d* configurations there arethree. There are none at all for ions with the d5 configuration. The energies of thetransitions in the former group depend only upon the strength of the crystal fieldthat is, upon the extent to which the metal electrons seek to avoid the ligands. Onthe other hand, the transition energies for ions in the second group are functions ofthe extents to which the d electrons avoid each other and the external environment.Quantitative considerations of both contributions yield much chemical information,as we shall see in Chapter 6. Our discussions throughout have been based on the crystal-field model. That isto say, the physical origin of the splittings has been ascribed to the repulsion of thed electrons by negatively charged ligands. Even at this early stage in our exposition,it is well to realize that the same qualitative picture would emerge whatever thedetailed physical origin of these splittings is, provided it is such that the d electronsclosest to ligand electron density are most raised in energy. In one sense, of course,that will suffice as a definition of repulsion, but there is much more to be said yetabout the origin of these effects. 3.12 Concluding Remarks 59

Suggestions for further reading

1. B.N. Figgis, Introduction to Ligand Fields, Wiley, New York, 1966.2. RA. Cotton, Chemical Applications of Group Theory, 3rd ed., Wiley, New York, 1990. - These references describe the material of chapter more fully and at a somewhat more technical level; they provide good insight.3. CJ. Ballhausen, Introduction to Ligand-Field Theory, McGraw-Hill, New York, 1962.4. J.S. Griffith, Theory of Transition Metal Ions, Cambridge University Press, Cambridge, 1961. - These two books delve much more deeply, and at a much more mathematical level, into this subject.5. M. Gerloch, Orbitals, Terms and States, Wiley, New York, 1986. - This book describes these quantities for atoms and linear molecules thoroughly but at a not-too-difficult mathematical level.4 The Intensities of 'd-d' Spectra

4.1 Transition moments

We shouldn't think of the absorption of light by a molecule in an anthropomorphic

way as if the molecule takes in the light and looks for something to do with it! Thevarious electronic states in a molecule correspond to discrete arrangements of theelectrons in the molecule. When we refer to a molecule as 'being in its ground stateor one of its excited states' we mean that it has such-and-such an electronicarrangement with such-and-such an energy. The absorption of light by a molecule isan interaction of the light with the molecule. The effect upon the molecule is thatits electrons rearrange from one state to another. We talk of a spectroscopic transitionfrom one electronic state to another. As mentioned in Section 2.1, the lifetime ofexcited electronic states is normally very short - say 10~18 sec - and the electronsspontaneously rearrange to the ground state with re-emission of the light. These tran-sitions only take place when the frequency (or energy, via the relationship E = hv)of the light matches the energy difference between the two states involved in thetransition. The probability of a given transition taking place depends upon the initialand final states and upon the transition-inducing properties of the light; where westart, where we finish, and what causes it. We define a transition moment, β, as theintegral,

Q=] V*gd ' (light operator) · ψ^ άτ

= < i//gd I light operator I ψ^> (4.1)

where i//gd and i//ex represent the ground and excited state wavef unctions. Theintensity, 7, of this transition - that is, its probability - is given by the square of thetransition moment,

/oc Q2 ^4 2 )

or by / oc Q* Q when Q is complex.

Now we need to know a little about the nature of the transition-inducing light.Light is a longitudinally propagating transverse oscillating electromagnetic field.As shown in Fig. 4-1, the electric and magnetic oscillations are at right angles toone another and to the direction of propagation. The arrows in the diagramemphasise that at any point along the propagating beam, there are electric and

magnetic fields of definite sign or parity (except, trivially, where they are zero). Inshort, these electric and magnetic fields are dipolar. We write our operator for thelight in Eq. (4.1), therefore, as a combination of electric and magnetic dipoleoperators. However, essentially because of the small size of molecules (or of thechromophoric part of the molecule - meaning that part undergoing electronictransitions) compared with the wavelength of the light typically used, the ability ofthe magnetic dipole to cause transitions is very slight - typically 10~4 times that ofthe electric dipole. Accordingly, we need not consider so-called magnetic-dipoletransitions any further and write transition moments as

Q =< (4.3)

where the electric-dipole operator er comprises the electronic charge, e, and theradius vector, r, describing the orientation of the unit light electric dipole.

4.2 Selection Rules

Suppose each state wavefunction in Eq. (4.3) can be written as a simple product ofspace-only and spin-only parts: then Q is given by

Q = < Wee Vspin I βή l/A'space (4.4)

where we have changed notation by using primes for the excited state fortypographical reasons. The space parts of the wavefunctions depend only uponordinary (x, y, z) space while the spin parts are functions only of spin space. Theoperator er is a function only of ordinary space and therefore does nothing to anyspin function. Regrouping variables within the integral, β, we find Eq. (4.5). 4.2 Selection Rules 63

Q = < Vspace ' er

' We > < Vspin I (4.5)

Regardless of the nature of the space parts, Q vanishes if y/spin^ V^'spm· ^ Q vanishes,so does /. Thus we have the so-called spin-selection rule which denies the possibi-lity of an electronic transition between states of different spin-multiplicity and wewrite AS = O for spin-allowed transitions. Expressed in different words, transitionsbetween states of different spin are not allowed because light has no spin propertiesand cannot, therefore, change the spin. Now consider a transition between states of the same spin. The 'spin overlapintegral', < i/Aspin I yA'spin>, in Eq. (4.5) is non-zero: if all relevant functions arenormalized, it is unity. So we turn our attention to the space part of Q (Eq. 4.6).

Q = < Vspace I er I i/A'space > ; space part (4.6)

Generally, such integrals are calculable only with great difficulty because werarely know the exact forms of the wavefunctions ψ. However, a great deal can beestablished simply by considerations of symmetry. We rely on a generalmathematical concept that integrals of odd functions vanish. An odd function is onewith two parts (or, in general, two sets of parts) which are identical in shape butopposite in sign. An even function is one with parts which are identical in shapeand sign. Figure 4-2 shows some examples. A particular, though not unique, subsetof such functions are those which are odd or even with respect to inversion througha centre of symmetry. Examples we are familiar with are the d and ρ orbitals, beingeven and odd respectively.

odd odd

even even

Figure 4-2. Examples of odd and even functions.

Let us enquire about the electric dipole transition moment between two d orbitalsas expressed in Eq. (4.7).64 4 The Intensities of 'd-d' Spectra

Gspace = < d I er IJ' > (4.7)

The d orbitals are centre symmetric and are of g symmetry. The light operator,being dipolar, is of u symmetry. The symmetry of the whole function under theintegral sign in (4.7) - that is, for the product d-r-d- is g X u X g, namely u. Theintegral over all volumes of a u function vanishes identically. Since Q in (4.7) thenvanishes, so does the intensity /. In short, 'd-d' transitions are disallowed. This is an example of the parity rule, also known as Laporte's rule: transitionsbetween orbitals or states of different parity are forbidden. It means that all d-d,p-p, s-s and/-/orbital transitions, for example, are forbidden. Actually, thedisallowedness of these transitions is underscored by another selection rule. Thephotons of light possess one unit of angular momentum (a spatial property). The ab-sorption of a photon during the course of a transition vectorially adds exactly one unitof angular momentum to the molecule. This means that the two orbitals in a tran-sition must have orbital angular momenta that differ by exactly one unit: Al = ±1.Note also that the orbital selection rule is Al = ±1 and NOT AL = ±1. The absorptionof one photon of light leads to the rearrangement of one electron, that is, we aredealing with a one-electron property here - hence the lower-case /. So transitions5- —> ρ, ρ —» d, d —>/are allowed but not d —» d etc.: the rule also forbids transitionslike s —> d or ρ —»/ which would be allowed by the parity rule.

4.3 ' Violation' of the Selection Rules

These selection rules appear to predict that transition-metal complex spectra willhave no 'd-d' bands. But, of course, they do! The rules are strictly absolute: theymay not be violated at all. Their apparent violation derives from the nature of theelectronic states involved in any transition. At the beginning of Section 4.2, wesupposed that the relevant wavefunctions could be factorized into space-only andspin-only parts. While that is nearly true, it isn't exactly so. The magnetic interactionwhich couples the orbital- and spin-angular momenta of electrons, that we call'spin-orbit coupling', means that neither spin nor orbital properties are constantwith time. In the same way that the electrostatic coupling between electrons causesthe angular momenta of each electron to vary with time in favour of constant spinand orbital momenta for the electrons as a group, the magnetic interaction yieldsonly a total angular momentum that is constant. For a single electron, we write thetotal angular momentum j as the vector sum in (4.8)

j =I+s (4.8)

and for a group of electrons, we write it as in (4.9).

J =L +S (4.9) 4.3 'Violation' of the Selection Rules 65

A term label like 3F, for example, is thus no longer strictly meaningful for it impliesconstant spin- and orbital angular momentum properties (S = 1, L = 3). Oneconsequence of spin-orbit coupling is a 'scrambling' of the two kinds of angularmomentum. So a nominal 3X term may really more properly be described as amixture of terms of different spin-multiplicity as, for example, in Eq. (4.10).

i3 X' = 3X or ' Q' = 4Q + b2R , etc 4 (4.10)

The mixing coefficients a and b in (4.10) depend upon the efficiency of the spin-orbit coupling process, parameterized by the so-called spin-orbit coupling coefficientλ (or ζ for a single electron). As λ—>0, so also do a or b. Spin-orbit couplingeffects, especially for the first period transition elements, are rather small comparedwith either Coulomb or crystal-field effects, so the mixing coefficients a or b aresmall. However, insofar that they are non-zero, we might write a transition momentas in Eq. (4.11).

The first two terms in the expansion are strictly zero because of the spin selectionrule, while the last two are non-zero, at least so far as the spin-selection rule isconcerned. So a 'spin-forbidden' transition like this, '3X'- > C 1 W , can be observedbecause the descriptions 3X and 1W are only approximate: that is why we enclosethem in quotation marks. To emphasize: the spin-orbit coupling coefficients for thefirst row transition elements are small, the mixing coefficients a and b are small,and hence the intensities of these spin-forbidden transitions are very weak. Consider now spin-allowed transitions. The parity and angular momentumselection rules forbid pure d <-> d transitions. Once again the rule is absolute. It isour description of the wavefunctions that is at fault. Suppose we enquire about a 'd-d' transition in a tetrahedral complex. It might be supposed that the parity ruleis inoperative here, since the tetrahedron has no centre of inversion to which the dorbitals and the light operator can be symmetry classified. But, this is not at alltrue; for two reasons, one being empirical (which is more of an observation than areason) and one theoretical. The empirical 'reason' is that if the parity rule wereirrelevant, the intensities of 'd-d' bands in tetrahedral molecules could be fullyallowed and as strong as those we observe in dyes, for example. In fact, the 'd-d"bands in tetrahedral species are perhaps two or three orders of magnitude weakerthan many fully allowed transitions. The theoretical reason is as follows. Although the placing of the ligands in atetrahedral molecule does not define a centre of symmetry, the d orbitals arenevertheless centro symmetric and the light operator is still of odd parity and sod-d transitions remain parity and orbitally (Al = ±1) forbidden. It is the nuclearcoordinates that fail to define a centre of inversion, while we are considering a66 4 The Intensities of 'd-d' Spectra

transition of electron coordinates. However, with respect to those nuclear

coordinates, no functions can be labelled as g or u. It is entirely possible thatproper wavefunctions (solutions of the Schrodinger equation for the molecule) mayinvolve mixtures of, say, d and ρ wavefunctions. A true wavefunction that we label4 J' (using the quotation mark convention again) might more properly be expressedas in Eq. (4.12).

'd' = d + cp (4.12)

Appropriate transition moments then take the form in Eq. (4.13).

Q = <'d' I er I 'd" >

The first two parts of the expression vanish exactly because of Laporte's rule,while the last two survive both parity and orbital selection rules to the extent thatthe mixing coefficients c and c' are non-zero in noncentric complexes. Experimentally, spin-allowed 'd-d' bands (we use the quotation marks again)are observed with intensities perhaps 100 times larger than spin-forbidden ones butstill a few orders of magnitude (say, two) less intense than fully allowed transitions.This weakness of the 'd-d' bands, alluded to in Chapter 2, is a most importantpointer to the character of the d orbitals in transition-metal complexes. It directlyimplies that the admixture between d and p metal functions is small. Now a ligandfunction can be expressed as a sum of metal-centred orbitals also (see Box 4-1).The weakness of the 'd-d' bands also implies that that portion of any ligandfunction which looks like a p orbital when expanded onto the metal is small also.Overall, therefore, the great extent to which 'd-d' bands do satisfy Laporte's ruleentirely supports our proposition in Chapter 2 that the d orbitals in Werner-typecomplexes are relatively well isolated (or decoupled or unmixed) from the valenceshell of s and/or p functions. Now look at octahedral complexes, or those with any other environment posses-sing a centre of symmetry (e.g. square-planar). These present a further problem.The process of 'violating' the parity rule is no longer available, for orbitals ofdifferent parity do not mix under a Hamiltonian for a centrosymmetric molecule.Here the nuclear arrangement requires the labelling of d functions as g and of pfunctions as u\ in centrosymmetric complexes, d orbitals do not mix with p orbitals.And yet 'd-d' transitions are observed in octahedral chromophores. We must turnto another mechanism. Actually this mechanism is operative for all chromophores,whether centrosymmetric or not. As we shall see, however, it is less effective thanthat described above and so wasn't mentioned there. For centrosymmetric systemsit's the only game in town. When discussing the origin of the large widths of 'd-d' bands in Chapter 2, wenoted that molecules are always vibrating. Some of these vibrations are such as toremove a centre of inversion. Consider just the one example in Fig. 4-3. This 4.3 'Violation' of the Selection Rules 67

Box 4-1Bonding orbitals in a metal complex may be thought of as molecular orbitals built fromappropriate metal and ligand functions. In the case of an M-L σ bond orbital, ψσ, forexample, we write

= CM Μσ CL Φσ

where M0. and Φσ are metal and ligand σ orbitals, and CM, CL are some mixing coefficients.It is usual to think of the metal orbital as centred upon (that is, expressed with respect to)the metal and of the ligand orbital as centred on the ligand. Diagramatically, this can berepresented as the following.

- - M L^>«O

Mn Ψσ

For many purposes, it is more convenient to express all functions with respectto just one origin - most usually the metal. The expansion theorem may be exploited toexpress any function as an (infinite) sum of convenient 'basis' functions. Here we write thefunction centred on the ligand as a linear combination of functions centred on the metal

The figure below illustrates how such a linear combination of metal orbitals (takensequentially in the diagram for heuristic reasons) can reproduce the orbital on anothercentre. M L

GXD 4 The Intensities of 'd-d' Spectra

Box 4-1 (Continued)

The point of this little diversion has been to show how a bond orbital like ψσ above can, quite generally, be thought of as some linear combination of just metal-centred orbitals. Now, ultimately, the metal d orbitals become admixed with a little of the bond orbital ψ and so within that final mixture we find both d and ρ type orbitals (others too, but we focus here just on the d-p mixing). If the whole MLN complex is centrosymmetric, all such d-p mixings cancel identically, but not otherwise. Now comes the point. When such d-p mixing survives, was the ρ orbital originally on the metal (e.g. a metal 4/?), or part of the ligand function Φ? Of course, we cannot say. However, knowing that Laporte's rule is so well obeyed in practice, means that such d-p mixing is small and hence, even if all the ρ character originated on the ligand, we conclude that the metal-ligand orbital mixing is small.

vibration mode of an octahedron involves the stretching of one bond together witha simultaneous contraction of the bond on the opposite side of the metal (the otherligands make minor movements also, as shown). During the course of the vibration,therefore, the nuclear arrangement lacks any centre of inversion. At any instantduring the vibration, d-p mixing can occur (other than at the trivial point of zerodistortion) and, in the manner of Eq. (4.13), a 'd-d' transition can become partiallyallowed. The reader may object that during the other half of the vibration, thenuclear displacements will be reversed, as shown on the right in Fig. 4-3, so thatthe orbital mixing may change from dz + cpz to dz-cpz. Yet this doesn't implycancellation of contributions to the transition moment or intensity. This is because,once again, the typical period of a vibration is about 10~13 sec, compared with theapproximately 10~18 sec lifetime of the electronic excitation. During the course of a

Figure 4-3. An ungerade vibration of an octahedron.

4.4 Intensity 'Stealing' 69

Box 4-2 Selection rules: a reminder. A mistake often made by those new to the subject is to say that 'The Laporte rule is irrelevant for tetrahedral complexes (say) because they lack a centre of symmetry and so the concept of parity is without meaning'. This is incorrect because the light operates not upon the nuclear coordninates but upon the electron coordinates which, for pure d or ρ wavefunctions, for example, have well-defined parity. The lack of a molecular inversion centre allows the mixing together of pure d and ρ (or/) orbitals: the result is the mixed parity of the orbitals and consequent non-zero transition moments. Furthermore, had the original statement been correct, we would have expected intensities of tetrahedral 'd-d' transitions to be fully allowed, which they are not.

single vibrational cycle of the nuclei, there is time for around 105 electronicexcitations! Put it another way. An incident beam of light encounters an ensembleof many (perhaps 1020) molecules which appear stationary but at all possible stagesof the vibrational cycle. For each encounter of light and molecule, an effectivelystatic d-p mixing is in place and a "d-d" absorption can occur. Overall, the'violation' of the parity rule in vibrating octahedral chromophores is less than instatic tetrahedral ones because the average degree of d-p mixing is less in thedynamic environment than in the static. Typically, 'd-d' bands for octahedralcomplexes are about ten to one hundred times weaker than those for tetrahedralcomplexes. The mechanism just described is often called 'vibronic' coupling.

4.4 Intensity 'Stealing'

Occasionally, some bands which might otherwise be expected to be weak are

observed to be quite strong. Two examples are shown in Fig. 4-4. The first shows 2the electronic spectrum of a solution containing [CoCl4] " ions in nitromethane. For 7this d system, we expect three spin-allowed transitions and these are observed atroughly 3500, 7000 and 14,000 cm^ 1 . They correspond (see Chapter 3) to theexcitations 4 A 2 —> 4T2, —> 4T^(F) and —»4T1(P) respectively. Note, however, that theband at 14,000 cm"1 comprises several sub-maxima. In part, they are assigned tocomponents of the 4T^1(P) term that arise due to spin-orbit coupling. At least onecomponent, however, is assigned to a spin-forbidden transition, 4A2-^2X(2G). Thedetails of the assignment are unimportant for us. The feature of particularinterest here is that the intensity of the spin-forbidden transition is comparable withthose of the spin-allowed transition. Other spin-forbidden transitions elswhere inthe spectrum are very weak indeed, as generally expected. Why is this particularspin-forbidden band so strong? Well, recall the process that leads to spin-forbiddenbands being seen at all. In Eq. (4.11), the allowedness of such transitions is pro-portional to the mixing between states of different spin angular momentum causedby spin-orbit coupling. The degree of such mixing is in turn proportional to thespin-orbit coupling coefficient which is quite small for the first row transition-metal70 4 The Intensities of 'd-d' Spectra

ions. However, it is also roughly inversely proportional to the energy separation of

the states being mixed in this way. Occasionally, and the tetrachlorocobaltate(n) ionpresents just such a case, spin quartet and doublet (here) terms are brought intoclose energetic proximity by the crystal field. The resulting scrambling of theseterms by spin-orbit coupling can then be very large indeed, and it is quite possiblethat the true wavefunctions can approach 50:50 quartet-doublet character. Underthese circumstances, the 'spin-forbidden' transition may acquire a much largerintensity than usual. Of course, the intensity gained in this way is at the expense ofthe intensity of the 'spin-allowed' transition since it is 'diluted', as it were, by the 4 4 42 2admixed doublet character. Between them, the A2-^r1(P)' and A 2 ^ X( G)'transitions possess some total intensity. As the spin-orbit induced mixing scramblesthose wavef unctions, the distribution of intensity between them becomes more equal.We say that the spin-forbidden transition acquires intensity by 'stealing' from thespin-allowed band. Actually, all spin-forbidden intensities arise in this way, for theprocess summarized in Eq. (4.11) is really the full story, but the expression 'intensitystealing' appears to be reserved for bands with 'unusually' high intensities due totheir close proximity to more intense transitions. The second example in Fig. 4-4 shows how a (spin-allowed or spin-forbidden)band lying close to a charge transfer band may acquire unusually high intensity. Weshall discuss charge-transfer bands more in Chapter 6. For the moment, we notethat they involve transitions between metal d orbitals and ligands, are often fullyallowed and hence intense. On occasion, the symmetry of a charge transfer state 4.5 'Two-Electron Jumps' 71

may differ from that of an energetically proximate d state in a way that can bematched by a molecular vibration. If so, the two states can become mixed and the'd-d' transition acquires extra intensity at the expense of the charge-transfer band.The mixing may be small but when the charge-transfer band is very intense, theaugmentation of the 'd-d' intensity in this way can be considerable.

4.5 'Two-Electron Jumps'

Sometimes, spin-allowed bands are much weaker than otherwise expected. Therecan be many reasons for this, most of which require more detailed analysis than weare able to present here. One particular case, however, can be discussed. It is wellillustrated by the spectra of octahedral cobalt(n) species, an example being shownin Fig. 4-5. Three spin-allowed transitions are expected for these d1 complexes,namely 4Tlg(F)^4T2g, ->4A2g, -^4T18(P) - see Chapter 3. The bands in Fig. 4-5 areso labelled. Note the weakness of the 4rlg—>4A2g transition. The situation is quitetypical of the spectra of octahedral cobalt(n) complexes. On occasion, the -^4A28transition barely appears as a weak shoulder on the —> 4 T ig (P) band and can bemissed. Why is this band so weak? We get the answer by looking at the d1correlation diagram, the spin-allowed part of which is shown in Fig. 4-6. Observehow the ground term 4Tig(F) in the weak field correlates with the strong-fieldconfiguration t^e\\ 4T28 and 4Tlg(P) with ΐ£β\\ and 4A2g with t238e4. At the strong-field limit, therefore, the transitions 4rlg(F)—>4Γ2£, —> 4 T i g (P) involve the promotionof one electron from the t2g subset to the eg subset. On the other hand, the transition4 Ti8(F)->4A2g correlates with the promotion of two electrons from the t2g to the egset. This is an example of a so-called 'two-electron jump'. It is intrinsically lessprobable than a one-electron jump and so the ->4A2g band is only weakly observed. Of course, in real systems, the relative contributions of Coulomb and crystal-field effects are such as to place chromophores somewhere inbetween the weak- 4 4and strong-field limits. In that case, a real T18(F) —> A2g transition is not a puretwo-electron jump, so that some intensity is observed.

Box 4-3 Absorption of one photon of light results in the relocation (with respect to space, spin or both) of one electron. It is possible, but extremely unlikely, that a second photon, together with its associated electronic rearrangement, can be absorbed before the ground state is re- acquired upon expulsion of a photon. It's unlikelyhood is because the lifetime of the excited state is typically only 10~ 18 seconds or so.

4.6 'Spin-Flip' Transitions

Here we comment on the shape of certain spin-forbidden bands. Though not strictlypart of the intensity story being discussed in this chapter, an understanding of so-called spin-flip transitions depends upon a perusal of correlation diagrams as didour discussion of two-electron jumps. A typical example of a spin-flip transition isshown inFig. 4-7. Unless totally obscured by a spin-allowed band, the spectra ofoctahedral nickel (π) complexes display a relatively sharp spike around 13,000cm -1 . The spike corresponds to a spin-forbidden transition and, on comparing bandareas, is not of unusual intensity for such a transition. It is so noticeable because itis so narrow - say 100 cm" 1 wide. It is broad compared with the 1-2 cm" 1 of free-ion line spectra but very narrow compared with the 2000-3000 cm" 1 of spin-allowed crystal-field bands. 4.6 'Spin-Flip' Transitions 73

10 15 20 1 3 ν/cm- xlO

Figure 4-7. Spectrum of a typical, octahedral nickel(n) complex.

We briefly discussed the origin of the band widths of crystal-field spectra in

Section 3.2. The broadening results from the way molecular vibrations affect groundand excited state energies differently. Sometimes, however, the response of groundand one or more excited states to bond length (and other vibrational) changes canbe similar. The variations of all terms arising from the d* configuration with Aoct areshown in Fig. 4-8. This is another Tanabe-Sugano diagram of the type that we sawearlier for the d2 configuration. Notice how the energies of the 1E8 and 1T2^ terms from 1D vary with Dq in verynearly the same way as does that of the ground 3A2g term. Because of this parallelism,the transition energy from 3A2^(3F) —» 1 E g ( 1D) hardly changes during the course ofany vibration that affects the magnitude of Dq. The transition is thus seen as a

Figure 4-8. A term energy diagram for d*.

74 4 The Intensities of 'd-d' Spectra

sharp feature. Notice also that the same is not true of other spin-forbiddentransitions. What is it about these particular spin-forbidden transitions that causesthis parallel energy behaviour? A consideration of a complete J8 correlation dia-gram helps provide the answer. The various electronic transitions we considerinvolve, in the strong-field limit, the rearrangement of electrons within or betweenthe t2g and eg subsets. Some transitions, however, do not involve any spatialrearrangement but only a spin change. A transition from a spin-triplet term (withS = 1 and two unpaired electrons) to a spin-singlet term (with S = O and no unpairedelectrons) can be achieved by reversing the sense of just one electron spin. If that isall that happens - that is, if the spatial distribution of the electrons remainsunchanged - we refer to the transition as a 'spin-flip' transition. Since the spatialarrangements in ground and excited states for such a transition are the same, theirresponses to variations in the crystal-field strength (a space-only property) are thesame, and the parallelism in diagrams like the one in Fig. 4-8 results. To emphasizethis simple idea, we note that the transition within the t2g subset is an example of a'spin-flip' transition. Note, once more, that not all spin-forbidden transitions involveonly spin changes, so not all are of the spin-flip type and not all, therefore, aresharp. Our example of a 'spin-flip' transition is indicated by the arrow in Fig. 4-8.

4.7 The Effects of Temperature Change

We have just discussed one aspect of the shapes of 'd-d' bands. For interest' sake,we finish this chapter with one more. It concerns the frequent, but not invariable,observation that band maxima may move somewhat towards the blue (higherfrequency) end of the spectrum as a sample is cooled. In Fig. 4-9 are sketchedpotential energy curves for the ground and an excited electronic state. The 'ladders'represent the fundamental and various harmonic vibrational states associated witheach electronic state. The vibrational states are typically separated by a (very) fewhundred wavenumbers. Most molecules occupy the lowest vibrational state of theground electronic state. One or more of the higher vibrational states are occupied toextents depending upon temperature and the Boltzmann distribution. Essentially nomolecules occupy the excited electronic states. Electronic transitions promotemolecules from members of the ground electronic state to members of the excitedone. Those promoted from the higher lying vibrational states absorb light of a lowerfrequency than those promoted from the lowest vibrational state. As a sample iscooled, less molecules occupy the higher vibrational states and so the averageelectronic transition energy increases: the band maximum moves towards the blue,as sketched in Fig. 4-10. The progressive removal of the lower-energy, transitionson cooling is referred to as the loss, or depletion, of 'hot bands'. The effect in Fig.4-10 is not always observed for it depends upon the relative lateral displacementsof the two potential wells in Fig. 4-9. Such variations are difficult, if not impossible,to calculate and hence predict. In addition to these possible blue shifts, there is a general rule that the intensitiesof the 'd-d' spectra of centrosymmetric molecules decrease with cooling while 4.7 The Effects of Temperature Change 75

those of acentric chromophores do not. This follows from our discussions in Section4.3 in which we noted that the origin of 'd-d' intensities is vibronic (i.e. dy-namically sourced) in centrosymmetric species but intrinsic (statically sourced) innon-centrosymmetric chromophores.

etc.

Y' 'hot bands'

Figure 4-9. Transitions occur from ground and vibrationally excited states of the groundelectronic state to various vibrational components of the electronically excited state.

high temp

Figure 4-10. The qualitative appearance of the spectral band energies corresponding to thetransitions in Fig. 4-9.76 4 The Intensities of 'd-d' Spectra

4.8 Summarizing Remarks

Crystal-field theory (and its successor, ligand-field theory, which we discuss in

Chapter 6) forms a significant, and indeed large, part of standard texts and teachingcourses in transition-metal chemistry for several interconnected reasons. It isremarkably successful at rationalizing a great body of spectroscopic and, as weshall see in the next chapter, magnetic data. It provides, at first level at least, abroadly accessible exercise in the exploitation of quantum mechanics and grouptheory. And, provided no questions of great detail or subtlety are put, it haspredictive power, too. Again, we shall see more of that in the next two chapters.For the most part, its success and appeal are to be laid at the door of symmetry. Thenumbers of transitions to be expected in 'd-d' spectra, and their patterns, aredetermined at root by symmetry in one guise or another. Matters with more chemicalimport inevitably involve factors of a continuously variable kind and when suchconsiderations are incorporated into the crystal-field idea, predictions inevitablybecome more qualitative and arguments more subtle. A little of all this is evident inour descriptions of the 'violation' of electric-dipole selection rules. The rules areclear and sharp. Nature circumvents them a little by rendering them somewhatirrelevant. It all has been to do with the character of the wavefunctions in realsystems, which are eigenfunctions of a, generally, complicated Hamiltonian. Thelabels we use are useful because they are approximately apt. However, because theyare approximate, the rules get broken. How do we know these labels are reasonablyapt? Because the rules are only just broken. The d orbitals, though much affected bythe molecular environment, tend to mind their own business.

- All these references discuss selection rules from various points of view.5 Spin and Magnetism

5.1 High-Spin and Low-Spin Configurations

In Chapter 3, we concentrated on the numbers and patterns of spin-allowed 'd-d'

transitions because, as we discussed in Chapter 4, they are usually more intense andobvious than the spin-forbidden ones. In fact, the perceived colours of mosttransition-metal complexes are determined by the spin-allowed 'd-d' bands. Implicitthroughout our discussions in Chapter 3 was an assumption that the spin-degeneracies of the crystal-field ground terms were the same as those of thecorresponding free ions. We are not referring here to the fact that crystal-field termsarising from a free-ion term carry the same spin label, for that is always true.Rather, we are addressing the contest between interelectron repulsions and the crystalfield. In the weak-field limit, the crystal-field ground term must be one of maximumspin-multiplicity because such is the case for the free ion itself and the free ion isthe ultimate limit of a weak crystal field. To see if any different result is possible,we must move towards the strong field. We begin at the strong-field limit itself. In Fig. 5-1 we represent possible strong-field ground configurations for dn ionsin octahedral symmetry. Consider each dn ion in turn. For dl the lowest energyorbital arrangement (strong-field configuration) is that housing the solitary electronin the lower-lying t2g orbital subset. For J2, it is similarly best to place both electronswithin the t2g subset: the same goes for the three electrons of d3. For d4 we have achoice: we can place all four electrons within the lower-lying t2g set and suffer thecrowding entailed in placing two electrons within the same orbital or we put threeelectrons in the t2g set and one in the eg set and suffer instead the promotion energyt2g—><?g; Aoct. The choice in any particular case depends upon the relative energypenalties incurred from interelectron repulsion or from the crystal field. In thecontext of the present discussion, the penalty from interelectron repulsion is oftencalled the pairing energy, P. Now, notice also, that all four electrons will takeparallel spins in the configuration t^e\ because of Hund's rule while two electronsmust pair up their spins in t2g. These two arrangements or configurations are calledhigh-spin or low-spin, as appropriate. We refer here to the net spin of an electronicarrangement, that is, to how many unpaired electrons there are. One may formalizethe result with the inequalities given in Eq. (5.1).

For the d5 configuration, we similarly have two choices corresponding to the

strong-field configurations t2g and t^e\ for low- and high-spin arrangementsrespectively. The reader may ask why we don't consider an intermediate-spinarrangement t{\e\. There is an 'all-or-nothing' reason. If Aoct is too great to allowthe promotion of one electron from t25g it is still too great to allow the doublepromotion to t^ej. Conversely, if the pairing energy is too great to allow the formingof one pair of electrons, it is too great to yield two pairs. No octahedral d5 ions areknown with an intermediate-spin arrangement. The same is not true for moleculeswith other symmetries, e.g. planar, but that is not at issue here. The rest of Fig. 5-1 is completed in like manner. The dn ions with η = 4,5,6,7offer two choices - high- or low-spin - while for η = 1,2,3,8,9 only one arrange-ment of lowest energy is possible. It is left as an exercise for the reader to constructa similar diagram for dn ions in a tetrahedral field and show that high- and low-spinchoices exist for η - 3,4,5,6 but not for η = 1,2,7,8,9. Few if any examples of low-spin tetrahedral dn complexes exist, however, because zltet, being only 4/9^1oct, is notusually sufficient to prevent the e - t2 promotion i.e. P > Atei always. For the octahedral case in Fig. 5-1, we include mention of the number of unpairedelectrons associated with each arrangement. For real molecules we could use this todetermine which configuration is lowest in energy - whether Aoct or P were thegreater - if only we had some experimental method of measuring the number ofunpaired electrons. There is such a method and it depends upon the interaction ofthese molecules with a magnetic field.

5.2 The Qualitative Origin of Paramagnetism

All substances interact with a magnetic field - there are no exceptions. Substancesmay be subdivided according to their manner of interaction with magnetic fields invarious ways. An old classification which is directly empirical and generally usefulis to group materials together which are either a) repelled, b) attracted or c) attractedvery strongly. Only the latter are commonly recognized in everyday life. They arecalled ferromagnets. The group is not large - soft iron, cobalt, a few other metalsand alloys, as well as a small number of special compounds. Their interaction withmagnetic fields is many orders of magnitude stronger than that of the materials ingroups a) or b). The origins of their ferromagnetic property lies in cooperativeinteractions between molecules or atoms and their study properly lies within therealm of physics. Fascinating though they are, we have no more to say about them. Substances in group a) are repelled by a magnetic field and are called diamagnets.Diamagnetism is a universal atomic (and, hence, molecular) property and isgenerally very small in magnitude. That a bar magnet actually does repel a piece ofpaper, for example, can only be demonstrated with rather delicate apparatus. Themuch smaller, though nevertheless extensive, group of materials that are weaklyattracted by a magnetic field define paramagnets. Many transition-metal compoundsfall into this class. Paramagnetism, when it is present, is generally larger (often80 5 Spin and Magnetism

much larger) than diamagnetism. So although diamagnetism is universal in atoms -

and hence molecules - it is almost invariably swamped by the paramagnetic effectwhen that is present. Even so, the attraction of copper sulphate crystals, which areparamagnetic, to a magnet can only be observed, once more, with delicate apparatus.Diamagnetism is the essence of chemical shifts in nuclear magnetic resonancespectroscopy, for example, and so is a very important topic elsewhere in chemistry.For our present area, however, diamagnetic effects are treated as corrections to anyparamagnetism. Our remarks on magnetism from now on are therefore confinedexclusively to the case of paramagnetism. It is convenient to begin with a classical picture. Paramagnetic substances areconsidered to comprise molecules with permanent magnetic dipole moments, m.They may be regarded here as miniature bar magnets. All discussions of bulkmagnetism concern molecules en masse - molecular ensembles. In the absence ofan applied field, a paramagnetic sample comprises molecules whose permanentmagnetic dipoles are oriented randomly, as indicated in Fig. 5-2a, because of theever present thermal agitation to which all molecular ensembles are subject. On ap-plication of an external magnetic field, these molecular magnets will tend to alignparallel to the field. We can safely neglect here any tendency of the molecular

a) b)

Figure 5-2. In the absence of an applied magnetic field a), the molecular magnetic dipoles arerandomly oriented; on application of an external field b), the dipoles tend to orientate parallelto the field.

magnets to align parallel because of their magnetic interaction with each other: thatis because the strength of the magnetic field from any one such magnet is tiny ascompared with the externally applied field (which could be thought of as arisingfrom 1030 or more such aligned molecular magnets). Once again, thermal agitationwill prevent their aligning perfectly (Fig. 5-2b). Any particular molecular magnetaligned exactly parallel to the field, B, will possess energy -m-B, while the energyof any such magnet aligned exactly antiparallel will be +m-B relative to their energybefore the application of the field. Molecular magnets oriented between these extre-mes will acquire energies between these limits. In Fig. 5-3, we represent the ther-mal distribution of an ensemble of molecular magnets in an applied field. Since, onaverage, the molecular magnets tend to align more with the field than against it, the 5.2 The Qualitative Origin of Paramagnetism 81

+m.B

mean energy in field

Figure 5.3. The classical picture: the energies of dipoles varies continuously from parallelalignment with the applied magnetic field (-m-B) to antiparallel (+m-B). On the right is shownthe distribution of molecules that results and the lower mean energy of the ensemble relativeto the field-free environment.

average energy of the ensemble is less in the presence of an external field than inits absence, as also indicated in the figure. The phenomenon of paramagnetism thusarises because the energy of an ensemble of molecular magnets decreases - theensemble acquires more stability - on application of a magnetic field: the sample isattracted by the field. It is attracted because, if otherwise unconstrained, the samplewill move from a place of no field into a field for, by doing so, its energy isreduced. The energy change involves a redistribution between that for molecularalignment and that for thermal agitation; there is no exchange of energy with themagnetic field itself. Any (very, very slight) warming of the sample on applicationof the field is soon dissipated, an effect whose reverse usage allows for the techniqueof adiabatic cooling (see Box 5-1)

Box 5-1 The technique of adiabatic cooling is used to achieve temperatures lower than can be ob- tained by the conventional techniques of immersing a sample in liquid helium under low pressure, a process which might cool a sample to around 1.6 K. To cool a sample further, one can proceed as follows. The method depends upon the sample being held within a paramagnetic container. During the conventional cooling process, the paramagnetic holder is held within a strong magnetic field. According to the usual Boltzmann statistics, more molecules of the container occupy the lower energy levels than the higher ones. When equilibrium has been achieved, the magnetic field is switched off. The split energy levels return to a degenerate condition and the distribution of the molecules of the paramagnetic container reverts to a situation like that on the extreme left of Fig. 5-3. This change in distribution requires an input of thermal energy. That energy is taken from the environment, including the sample. Cooling of samples down to millikelvin levels can be achieved in this way.82 5 Spin and Magnetism

The greater the magnitude of the applied field, the greater the energy differencebetween parallel and antiparallel alignment of the molecular magnets and the lessable is the thermal agitation to randomize the molecular orientations. So, as shownon the right in Fig. 5-3, the mean energy of the ensemble decreases with increasingmagnetic field strength. This effect is exploited in various 'force technique' methodsof measuring magnetism. The Gouy method is illustrated in Fig. 5-4. The sample istaken in the form of a cylindrical rod - or, in the case of powders or solutions,contained within a cylindrical glass tube - to make integrations trivial. The sampleis suspended from a chemical balance, or any other force-measuring device, andplaced so that its bottom end lies near the strongest part of a magnetic field and its

Figure 5-4. Schematic arrangement for the Gouy

technique. The sample must be placed asymmetrically (vertically) in the magnetic field.

top end near the weakest part. The source of the magnetic field is convenientlyprovided by an electromagnet (or permanent magnet). Suppose the sample isbalanced in the absence of the magnetic field. On switching on the electromagnet,(or on introducing the permanent magnet) the sample will move toward the strongerpart of the field because more of the sample will enter a strong field and the energyloss, as in Figure 5-3, will be greater. On re-taring the balance, we observe thesample to weigh more in the field than out of it. In SI units, the force, F, on thecylindrical sample of cross-sectional area, a, is

(5-2) 5.2 The Qualitative Origin of Paramagnetism 83

where H is the strength of the magnetic field at the bottom of the sample and χ isthe mean volume susceptibility of the sample (see Box 5-3 for units). We now modify the classical picture set out above to accommodate quantummechanics.* Instead of talking about permanent molecular magnetic dipoles, wepostulate (for the moment, but we shall return to the point shortly) that paramagneticmolecules are those whose occupied energy levels (usually the ground or near-ground levels) possess a degree of degeneracy that is removed by the application ofa magnetic field. (We discuss an exception to this under 'temperature independentparamagnetism', later). Instead of Fig. 5-3, we construct Fig. 5-5. The splitting ofthe levels (a) in a magnetic field is symmetrical (b), as it was in the classicalpicture. The distribution of molecules amongst the component states is no longerequal in the presence of the applied field. An appropriate (Boltzmann) populationhistogram for the distribution of molecules amongst these states is shown in (c).Once again, we see that the mean energy of the molecular ensemble is less afterapplication of the field than before it and the phenomenon of paramagnetism followsonce more.

(a) (b) (c)

O -

B7 > B

Figure 5-5. The quantum mechanical picture: discrete population histograms take the place ofcontinuous distributions. The overall paramagnetism increases with increasing field strength.

* A careful classical analysis of all magnetization phenomena shows them to vanish identically!The interactions of matter with magnetic fields that we observe on a day-to-day basis arepurely quantum phenomena just as the existence of magnetic fields is a relativistic phenomenon.The classical prediction of vanishing magnetism is really as great a failure of the classicalregime as the better-known 'Ultraviolet catastrophe'.84 5 Spin and Magnetism

Also illustrated in Fig. 5-5, is the greater splitting (d) that follows an increasedmagnetic-field strength, whose splitting is, in fact, linear in the field strength, B.The Boltzmann distribution changes accordingly to (e), the mean energy decreases,and the force on the sample in a Gouy experiment increases. Quantitative analysisshows that the force on the sample varies linearly with the field strength until thatfield grows very large when the effect falls off and one observes (theoretically andexperimentally) the phenomenon of magnetic saturation, as shown in Fig. 5-6. Undernormal laboratory conditions, magnetic saturation is rarely observed.

slope = susceptibility

saturation

Figure 5-6. Saturation and the definition of magnetic susceptibility, χ.

Box 5-2 Classically, 'saturation' occurs when the field is so strong and/or the thermal agitation (temperature) is so feeble that all the molecular dipoles are aligned with the field. Then, of course, increasing the applied field more is unable to cause any further alignment.

Rather than quote some (mass normalized) force on the sample at each of severalfield strengths, it is sufficient to report the slope of the linear part of the curve inFig. 5-6. This slope is called the magnetic susceptibility of the sample. Units forsusceptibility, χ , and related quantities to be discussed in this section are reviewedin Box 5-3. The effect of temperature upon the situation in Fig. 5-5 is to modify theBoltzmann distribution. Lowering the temperature depopulates the higher-lyingenergy levels in favour of the lower. Therefore, susceptibility increases withdecreasing temperature. Quantitative studies of the simple (first-order*)

*'First-order' means that we consider nothing beyond that described here. In 'second-order',we would include the effects of mixing between ground and excited states brought about by themagnetic field. This is briefly discussed under 'second-order Zeeman effects' later. 5.2 The Qualitative Origin of Paramagnetism 85

Box 5-3 Units for magnetochemical quantities

B = H + 4πΜ : CGS

B and H can be quoted in gauss

B = μ0(Η + M) : SI

B and H are quoted in tesla (T): IT = 104 gauss

Several texts describe B as the applied magnetic field and H as the field in the sample. This is incorrect since both B and H exist inside and outside the sample. B is the magnetic field associated with a current loop source. H is the magnetic field associated with a (fictitious) magnetic point monopole: see Magnetism and Ligand-Field Analysis by M. Gerloch (Cambridge University Press, 1983). B and H differ very little with respect to magnitude or direction for weakly magnetic (non-ferromagnetic) materials. Many confusions about B and H arise because of this.

BIH = /I0(I +X) : SI

Β/Η = I + 4πχ : CGS

where χ is the volume susceptibility (dimensionless). Gram susceptibility, χ8, is defined by

xg = χ/Ρ where ρ is the density of the sample. Molar susceptibility, χΜ, is defined by

XM = XgM

where M is the molecular weight of the material. In the SI system, χΜ is measured in

HI3ITiOl"1; in the CGS system it is measured in cm3 mol"1. To convert χΜ values quoted in the CGS system into SI values, multiply by 4π χ ΙΟ"6. Effective magnetic moments, /zeff, defined in (5.4), are quoted in Bohr magnetons in either SI or CGS systems. In the CGS system the Bohr magneton is 0.92731 χ 10~20 erg gauss'1 whilst in the SI system it is 0.92731 χ 10~23 A m2 molecule'1. The magnetic moment, ^, is then 2.8279(^M T)1/2 B.M. in the CGS system, and 7.9774 χ 102(χΜ T)172 B.M. in SI. These expressions yield the same numerical values for μ^, so that expressions like (5.6), (5.7), (5.10) and (5.11) etc. remain valid in both CGS and SI systems.

circumstances of Fig. 5-5 predict that paramagnetic susceptibilities are inversely

proportional to temperature. In the later years of the last century, Pierre Curiesummarized a wealth of experimentation on paramagnetic substances with the lawthat bears his name:

Curie's Law χ = CfT (5.3)

where C is called Curie's constant. As a plot of χ versus l/T is a straight line,

according to this law and first-order theory, there is no need to report χ values atdifferent temperatures (see Fig. 5-7). It is sufficient to report the slope of such a86 5 Spin and Magnetism

1/λ Heff straight line

straighty hyperbola line

T T T

Figure 5-7. Curie Law behaviour of χ, Ι/χ and μ^ with respect to temperature.

relationship, and that slope is just C. For historical reasons we need not pursue, it isconventional instead to report a quantity known as the effective magnetic moment,jUeff, for a sample where μ&ίί is proportional to the square root of C: _ en (5.4) 2 mewhich is approximately the same as the relationship in Eq. (5.5).

Ueff = 2.828- (5.5)

Insofar that Curie's law is true, /ieff is independent of temperature, for that is howwe arrived at Eq. (5.5). In practice, Curie's law is rarely obeyed exactly and,occasionally, it is quite seriously flouted. Nevertheless it is still conventional toquote /ieff values but it is then necessary to quote them over a range of temperatures.Although we might just as well report susceptibility values in these circumstances,conventions die hard. In any case, the temperature variation for ^eff immediatelyand transparently reveals any departures from Curie's Law in a way that thetemperature variation of susceptibilities might not. We have seen how the phenomenon of paramagnetism follows from theassumption that paramagnetic molecules possess appropriate degenerate states whichsplit in a magnetic field. The question arises as to 'what molecular property leads tothese degeneracies?' and hence 'what does a magnetic susceptibility ultimatelymeasure?'. The answer is that paramagnetic moments or susceptibilities are ameasure of angular momentum - both spin- and orbital-angular momentum or thetotal angular momentum when that is a more appropriate quantity (i.e. when theeffects of spin-orbit coupling are large). Physically, and in outline only, it is simpleto see why this should be so. All measurements involve an interaction between thesystem and the apparatus. Interactions only take place between quantities of thesame kind. Orbiting or spinning electrons generate magnetic fields; applied magneticfields are generated by circulating electrons (current loops). The angular momenta of atoms are described by the quantum numbers L, S or /.When spin-orbit coupling is important, it is the total angular momentum / which isa constant of the system. A group of atomic wavefunctions with a common / value- akin to a term, as described in Section 3.6 - comprise (2J + 1) members with Mj 5.2 The Qualitative Origin of Paramagnetism 87

levels

Figure 5-8. Spin-orbit coupling splits the 3F term into three levels j. An externally appliedmagnetic field splits up the levels into their Mj components.

values ranging / , J-I,... -J. The (2 J + 1) degeneracy of such a level, as it is called,

is removed completely by an external magnetic field, as shown in Fig. 5-8. Theresult, as we have seen, is paramagnetism. Quantitative analysis of these circum-stances yields the formula in Eq. (5.6)

Meff = (5.6)

where g is defined as in Eq. (5-7).

5(5 + I)-L(L + !) = 1+ (5.7) 2 J(J + 1)

The L and S values are those from which the / value was formed via the vectorcoupling rule.* These formulae strictly apply only for the magnetism of free-ionlevels. They provide a good aproximation for the magnetism of lanthanidecomplexes, as we shall note in Chapter 10, but provide no useful account of themagnetic properties of d block compounds.A corresponding formula (Eq. 5.8), due to Van Vleck, has been derived for freeatoms in which the effects of spin-orbit coupling can be ignored.

(5.8)

Again, however, this is strictly applicable only for free ions. Even though spin-orbitcoupling is much less important for the first row of the d block, this formulaprovides a far less good approximation for d -block complexes than Eq. (5.6) doesfor lanthanide complexes. The reason is that the ground, and other, terms in these dcomplexes differ grossly from those of the corresponding free ion. These differencesare one result of the crystal field.

*(see Orbitals, Terms and States', Ch. 4).

88 5 5pm and Magnetism

5.3 'Orbital Quenching' and the 6SpIn-OnIy' Formula

We have nearly made contact again with our crystal-field discussions in Section5.1. In order that magnetic measurements be useful to us, however, we need tostudy the angular momenta associated with various crystal-field terms. First, werecall that no crystal field affects the spin angular momentum of any given free-ionterm.* As we observed in Section 3.7, crystal fields act only upon the spatial partsof wavefunctions. If a parent free-ion term has a spin-degeneracy (IS + 1), so alsodo the crystal-field terms that derive from it. Thus, any changes in angular moment-um brought about by a crystal field concern only the orbital (spatial) part. Consider the orbital angular momentum of a free-ion 3F term. Here L = 3 and theorbital degeneracy is 7. Application of Van Vleck's formula (5.8) predicts aneffective magnetic moment,

In octahedral symmetry, the F term splits into A2g + T2g + Tig crystal-field terms.Suppose we take the case for an octahedral nickel(n) complex. The ground term is3 A2g. The total degeneracy of this term is 3 from the spin-multiplicity. Since an Aterm is orbitally (spatially) non-degenerate, we can assign a fictitious Leff value forthis of O because 2L eff +1 = 1. We might employ Van Vleck's formula now in theform

(5.10)

Box 5-4 Strictly, L is defined only as a quantum number for a spherical environment - the free ion. The use of Leff = O for A terms or Lei{ = 1 for T terms on the grounds that (2Leff +1) equals the degeneracy of these terms is, however, legitimate as used here. There is a close parallel between the quantum mechanics of T terms in octahedral or tetrahedral symmetry on the one hand, and of P terms in spherical symmetry on the other.

3and so predict a magnetic moment, μ& = Λ/8. In other words, for this A2^ groundterm, the crystal field has completely removed the orbital angular momentum. Wesay that the orbital angular momentum has been quenched. Now take the case for an octahedral vanadium(m) ion. For J2, the ground term is3 Tlg. The spatial degeneracy of a Γ term is three-fold and we describe this with Leff= 1. Using (5.10) we find μ& = V l O . So for this 3 r lg term, the crystal field hasquenched some, but not all, of the angular momentum of the parent free ion F term.

* Except, apparently, insofar that for some dn configurations a strong crystal field may bringabout a 'low-spin' configuration as described in Section 5.1. However, in these cases, thecorresponding ligand-field terms correlate with excited free-ion terms which still have thesame spin-multiplicity as that of the strong-field term. 5.3 'Orbital Quenching' and the 'Spin-Only' Formula 89

Analogous arguments apply to the various ground terms of octahedral or tetrahedral

d2, d3, d1 and J8 complexes. The ground term for octahedral or tetrahedral d5 complexes is orbitally non-degenerate ( 6 A 1 ^ or 6Ai respectively) and so, once again, we expect no orbitalcontribution to the magnetic moment. The situations for J1, d4, d 6 and d9 are specialin one respect. When the ground term is T2g or T2, we have Leff = 1 and partialquenching as before. When, however, it is an E8 or E term, our simple ploy of usingLeff doesn't work and, in fact, these terms give rise to no orbital contribution at all(see Section 5.5). They are known as non-magnetic doublets because of this. Bewareof this jargon, by the way, for the lack of magnetism only refers to the orbitalcontribution; magnetism still arises from the spin angular momentum of 2E(g) or 5E(g)terms. Some explanation for the lack of orbital angular momentum for these cubic-field E terms will be given shortly. Meanwhile, we note that the phenomenon appliesonly to these E terms arising in strict octahedral or tetrahedral symmetry. Lowersymmetry environments also define E terms on occasion, but these are not generallybereft of orbital angular momentum. Altogether then, Van Vleck's formula for free ions is inappropriate for theparamagetism of crystal-field terms. Crystal fields partly or completely quench theorbital angular momentum (spatial degeneracy). To the extent that such quenchingis complete, we might consider using the limiting case of Van Vleck's equationwhere the magnetism is ascribed to the spin-angular momentum alone. This yieldsthe so-called spin-only formula (Eq. 5.11).

Ao = ^l 4S(S +Ϊ) (5.11)

Now the total spin-angular momentum quantum number S is given by the number,η , of unpaired electrons times the spin angular momentum quantum number s forthe electron, that is, S = n/2. Substitution of this relationship into Eq. (5.11) yieldsan alternative form of the spin-only formula,

(5.12)

which directly expresses the effective magnetic moment in terms of the number ofunpaired electrons. Of course, as discussed above, there may be some orbitalcontribution in any particular complex, but to the extent that the spin-only formulais appropriate, it provides the measure of the number of unpaired electrons that werequired at the end of Section 5.1.

By way of example, a d5 iron(m) complex with a magnetic moment close to 1.73

must, by reference to Fig. 5-1, be low-spin with Zioct >P since an iron(m) complexwith Zioct < P would have a magnetic moment of 5.92.90 5 5pm and Magnetism

5.4 Orbital Contributions

We have seen that there are orbital contributions to the magnetic moments ofcomplex ions with T ground terms but not with A or E terms. These rules are onlyapproximate, for the quantitative theory of paramagnetism is a rather complicatedaffair that we can do no more than skim in this book. One group of refinements toour earlier statements recognizes the over simplistic description of the true molecularwavefunctions. We saw the same sort of thing in our discussion of electric-dipoleselection rules in Section 4.3. Although interelectron repulsion and crystal-fieldenergies are much greater than spin-orbit coupling energies, the latter cannot beignored when we look at properties as sensitive as 'd-d' intensities or magneticsusceptibilities. Let us, therefore, take a second look at the ground wavefunctionsof octahedral J8 ions. Ignoring spin-orbit coupling, it is exact to label these ground wavefunctions withthe term label 3A2g, and it is equally exact to use the spin-only formula for themagnetic moment. Strictly, however, we should not ignore spin-orbit coupling for itcauses some mixing between the 3A2^ ground term and, for example, the higher-lying 3T2g term (Eq. 5.13).

VA( 4 3 A 2 /) = i/A(3A2g) + c ν/(3Γ2§) (5.13)

The mixing has nothing to do with the possibility of any molecules populatingthe 3T2g term, which is typically 12,000 cm"1 above the ground state term. Thepopulation of such a term is of the order ^-12000/20° at room temperature (kT ~ 200cm"1 at 300 K), which is absolutely negligible. The mixing arises because adescription of the molecular Hamiltonian in terms of Eq. (5.14) is incomplete andshould be replaced with Eq. (5.15).

(5.15)

3Wavefunctions like ψ( Α2§) are eigenfunctions of ^ί\ but those of 0-C2 are slightly ί3different. In writing them like ψ( Α2§') we merely indicate what the wave functionsare most nearly like. The functions i//t' 3 A 2g ') can be expressed in literally an infini-te variety of ways, although these must be explicitly determined. One way which israther convenient for further discussion is that shown in Eq. (5.13). In words, wemight say that 'under spin-orbit coupling, the ' 3 A 2 / term looks as if it containssome 3T2g character'. The extent of the mixing - the magnitude of c in (5.13) - is proportional to thecause of the mixing and inversely proportional to the energy separation of the 5.4 Orbital Contributions 91

original terms being mixed, that is, c ^ λ/Δοοι in this case. The admixture of Tcharacter into the formal A ground term implies admixture of orbital angularmomentum. Detailed theory yields the expression,

A4ff = Mso(l-4AM oct ) (5.16)

as the result of this on the paramagnetic moment. For an octahedral nickel(n)

complex, for example, Zi0Ct is typically 12,000 cm'1, λ is -315 cnr1 (negative becaused* is a more-than-half-filled shell), /iso = 2.83 and so, from Eq. (5.16), we calculateμ&ίί = 3.13. This is, indeed, the sort of value that is typically observed for suchcomplexes. For octahedral chromium(m) complexes, d3, with formal 4A2g groundterms, the less-than-half-filled d shell means λ is positive - though rather less thanthat for ds in magnitude - and Eq. (5.16) predicts magnetic moments somewhat lessthan that given by the spin-only formula, and that is also observed in practice. Theformula in Eq. (5.16) applies to all cubic-field systems having an A2g ground term.

and λ takes positive values for a less-than-half-filled shell, and negative values for a more- than-half-filled shell. These signs conform with Hund's third rule that minimum / values lie lowest in energy for less-than-half-filled shells, and highest in energy for more-than- half-filled ones.

Ion Ti3+ V 3+ Cr3+ Mn3+ Fe2+ Co2+ Ni2+ Cu2+

A similar expression has been derived for cubic-field complex ions having an Eground term (Eq. 5.17).

Meff = Ms0(I -2AM0Ct) (5.17)

The basis for this formula is just the same as described above but, in this case,spin-orbit coupling admixes the higher-lying J2(g) term wavefunctions into the groundE(g). The coefficient 2 in Eq. (5.17) rather than the 4 in Eq. (5.16) arises from thedifferent natures of the wavefunctions being mixed together. These two formulae describe orbital contributions to ground A or E terms thatarise by so-called second-order spin-orbit coupling with appropriate excited92 5 Spin and Magnetism

wavefunctions. For complexes with Γ ground terms, the effects of spin-orbit couplingare first-order and rather complicated. We describe them in barest outline only.Recall how a free-ion 3F term splits under spin-orbit coupling to give three so-called levels, 3F4, 3F3, 3F2, with total angular momenta / values of 4, 3 and 2respectively. In a similar way, a 3Tlg crystal-field term, for example, splits into threecomponents with J values of 2, 1 and O (we use the vector coupling rule with S=Iand Leff =1). The energy separations between these components are typically of theorder 50-500 cm"1 for first row transition-metal complexes and so there aresignificant molecular populations of each of them. Each component generally splitsup in an applied magnetic field and gives rise to a contribution to the magneticmoment which depends, in part, upon the population of that component. Thisimmediately tells us that the magnetic moments of such systems are generallytemperature dependent, at least because the populations of the original componentsare temperature dependent. In addition to this complex behaviour is the second-order Zeeman effect. Thesplitting of a degenerate set of wavefunctions by an applied magnetic field, asillustrated in Fig. 5-5, is called a first-order Zeeman splitting. Consideration of thiseffect alone always yields a Curie-Law behaviour and temperature-independentmagnetic moments. However, spin-orbit coupling is not the only mechanism thatcan scramble wavefunctions. The applied magnetic field itself effectively polarizesthe wavefunctions as well as changing their energies in such a way that any onewavefunction looks as if it has some of the character of (most) other wavefunctionsadmixed. This is the second-order Zeeman effect. (In truth, there is only one Zeemaneffect and these names merely refer to two parts of a mathematical expansion). Theamount of such magnetic-field-induced mixing is (again) proportional to the cause -the magnetic moment operator - and inversely proportional to the energy separationsof the wavefunctions being admixed. For typical laboratory magnetic fields, thenumerators here are of the order 0.1-1 cm"1 so the second-order Zeeman effectsare often very small indeed. However, in the case of an ion with a formal orbitaltriplet ground term split into fairly close-lying spin-orbit components, these effectscan be quite important. This is because the mixing induced by the magnetic field,though proportional to the small magnetic-field term (small numerator), is inverselyproportional here to a fairly small energy denominator. They can be calculated, buta discussion of how is outside our present scope. Suffice it to say that the magneticmoments of ions with formal orbital triplet ground terms have orbital contributionswhich are not simply estimable by the tactic of using Leff = 1 as we describedearlier. Furthermore, they tend to vary considerably with temperature. For ions with formal orbital singlet ground terms, it is often quite adequate toignore second-order Zeeman terms since any magnetic-field-admixed wavefunctionsare energetically well removed from the ground state. There is one type of situation,however, when these small effects are observable because they are the onlycontribution to magnetic susceptibilities. The classic case is that of low-spinoctahedral cobalt(m) complexes. From Fig. 5-1, we note the ground strong-fieldconfiguration to be t2g. The subshell is full and hence uniquely defined. There is nospin- or space-degeneracy associated with this electronic arrangement and it is 1labelled A 1 ^. Having no degeneracy at all, it cannot be split by an applied magnetic 5.5 Orbital Contributions at the Strong-Field Limit 93

field and so we expect no paramagnetism, just the diamagnetism that is a property

of all atoms. Experimentally, however, such systems are observed to be slightlyparamagnetic with magnetic moments of about 0.5 BM that vary with temperatureaccording to a square-root relationship, μβίί ^ Vr. Actually, this temperaturedependence of μβίί is misleading because of the way in which μ& is defined in Eq.(5.4). It conceals the more interesting fact that the susceptibility is independent oftemperature. Indeed, the phenomenon we describe here is called temperature-inde-pendent paramagnetism (T.I.R) Its origin is as follows. The ground state is indeed non-magnetic as discussed above, but only in firstorder. That is, there is no first-order splitting - no first-order Zeeman effect. Thereis however, a second-order Zeeman effect in which the applied magnetic field mixesvarious excited-state character into the ground state. Some of that excited-statecharacter arises from degenerate (paramagnetic) states. As a result of the mixing,the ground 1 A 1 ^ state remains non-degenerate (and so cannot split) but decreases inenergy by an amount proportional to the magnetic perturbation squared and inverselyproportional to the energy separation between the admixed states. All the moleculespopulate this unique ground state and the system has acquired a lower energy byexposure to the applied field. It is therefore paramagnetic. However, because theground state is unique there can be no change of thermal distribution amongstlevels as the temperature is changed and so the paramagnetic susceptibility is inde-pendent of temperature.

5.5 Orbital Contributions at the Strong-Field Limit

Our discussions of orbital contributions to magnetic moments began with the sim-ple rules for A, E and T terms in the weak-field limit. Analogous rules can beconstructed when we consider ions in terms of their strong-field configurations. Wealready had an example with the t26g configuration of octahedral cobalt(m) above.An orbital contribution will be made when there is an orbital degeneracy. In theconfigurations t2], t2j, t2g, t25g there exists the respective three-fold spatial degeneraciesgiven below.

For the cubic-field (octahedral or tetrahedral) subshells £(g)i there is spatial

degeneracy for e($ but not for e(£}. Nevertheless, neither of these configurations giverise to an orbital contribution to the magnetic moment. The conditions for orbitalcontributions to arise in strong-field configurations are that the orbitals must be94 5 Spin and Magnetism

Box 5-8 Orbital angular momentum associated with the dxz and dyz orbitals as a pair. The orbitals dxz and dyz can be expressed in terms of the complex forms ^1 and d_\ whose 2 2 angular parts are given by the spherical harmonics T1 and F_! , respectively. The matrix of orbital angular momentum about the z axis in the complex basis is

and we observe the obvious result that d{ and d^ orbitals have ±1 unit of orbital angular momentum about the ζ axis. In the real orbital basis, where

dl = rf_, - rf.) (U)

the equivalent matrix under /z is

d d xz yz (Ui)

On diagonalization, we find the real d orbitals to possess ±1 unit of orbital angular

momentum about z, when taken as a pair. Although the matrices (i) and (iii) tell the same story, one can barely draw the complex orbitals d±i yet their angular momentum is obvious. If we prefer to use the real forms in (ii), which we can draw, the orbital angular momentum is hidden in the imaginary off-diagonal elements of (iii). These off-diagonal elements have the form

<dxz\ I1 \dyz> = <dxz\-i\dxz> (iv)

or, in other words, /z rotates dyz into dxz (and multiplies it by -/). Now the combinations (ii) (or, conversely, d±i from dxzt dyz) can only be constructed if they are degenerate. All these features, fully discussed for px and py orbitals in Orbitals, Terms and States', are encapsulated within the rules given in the main text above.

degenerate, that the degenerate subsets must be neither full, empty nor exactly half-full, and that at least two of the orbitals within a subset must be related by rotationabout the z axis. Within the t2 subset, dxz becomes dyz on rotation about z and sogives rise to an orbital contribution. On the other hand, dz2 and dx2_y2 of the e subsetare not related by rotation about z and give no such orbital contribution. They form 5.6 The Chemical Relevance of Departures from the Spin-Only Formula 95

the so-called 'non-magnetic doublet' discussed earlier. Although these conditions

for orbital magnetic contributions are consistent with the summary given earlier inthis section, they must surely seem no more than a magical recipe. For those familiarwith the real and complex forms of wavefunctions, a more satisfactory explanationof these rules is given in Box 5-8. Exemplifying the use of the rules in applicationto the strong-field octahedral configurations of Fig. 5-1, for example, we note thatorbital contributions are expected for dl, J2, low-spin J4, low spin J5, high-spin J6,low-spin d7, and d9 configurations.

5.6 The Chemical Relevance of Departures

from the Spin-Only Formula

Careful and detailed studies of 'd-d' spectra and magnetic susceptibilities, prefer-ably on samples in the form of single crystals, can yield considerable insight intothe bonding in transition-metal complexes. One thinks here of the various specialisttechniques that are part of contemporary research which, of course, form no part ofour brief in this book. It is the case, however, that our somewhat technicaldiscussions of orbital contributions, second-order Zeeman effects and the like, areof direct relevance to the simple exploitation of paramagnetism in inorganicchemistry. They were recognized to be such even in the early days of crystal-fieldtheory and magnetochemistry. As we have seen, magnetism may be exploited - strictly through the spin-onlyformula - to count the number of unpaired electrons in a complex and thence toinfer something of the nature of the bonding in that complex. This early idea is stillemployed today at a 'finger printing' level. Within the context of our openingremarks in Section 5.1, a count of unpaired spin can differentiate between strong-field and weak-field environments. These unpaired-electron counting games canonly work so simply, however, if we have a strong correlation between the measuredproperty of susceptibility (or effective magnetic moment) and the number ofunpaired electrons. The 'spin-only' formula in Eq. (5.11) offers that simpleconnection. Like so much else in chemical theory, however, it only works when itworks! Our discussions in the past few sections show that the formula is sometimesexcellent, often reasonably accurate, but just as often inadequate. As such, thesimplicity offered by the spin-only relationship is spoilt. Since we do understandwhy and when, however, as described briefly above, simplicity may have been lostbut not understanding. These remarks are made, therefore, to assure the reader that connections betweenthe number of unpaired electrons in a complex and its magnetic properties - andindeed much more detail - are perfectly possible and well understood, notwith-standing the necessarily brief review of the subject that has been possible in thepresent non-specialist text.96 5 Spin and Magnetism

5.7 Summary

In crystal fields of any symmetry, ambiguities can, but need not, arise in assigningthe spin-degeneracy of the ground configuration. Such ambiguities arise from theconflict between the desire of electrons to avoid each other versus their desire toavoid negatively charged regions in the environment. In octahedral species, thespin-degeneracy is determined by the relative size of Aoct and the mean pairingenergy P. The crystal-field splitting in tetrahedral complexes of the first rowtransition-metal complexes is never greater than the pairing energy so that,empirically, only high-spin tetrahedral complexes are observed. The spin-degeneracyof actual systems may be determined from measurements of the magnetic moment.The simplest relationship between spin and magnetism is the limiting case of thespin-only formula relating magnetic moment directly to the number of unpairedelectrons. More careful scrutiny of crystal fields and magnetism provides a 'secondtier' of sophistication in which departures from the 'spin-only' formula can beanticipated. Orbital contributions to magnetism - which is the antithesis of orbitalquenching - are of second order for ions with A or E ground terms, but of firstorder and complicated for T ground terms. Once again, these qualitative rules deriveessentially from symmetry. The details of the physical nature and origins of crystal-field splittings are irrelevant for their establishment. Similar remarks have beenmade at the conclusions of each of the last three chapters. It is time now to come togrips with the more quantitative side of crystal-field theory and to correlate itssuccesses with other notions of chemical bonding.

6.1 The Nephelauxetic Effect

We saw in Chapter 3 how three spin-allowed transitions arise in octahedral or

tetrahedral complexes of metals with J2, d3, d1 or d8 configurations. We also learnedthat the energies of those transitions depend upon the magnitudes of the crystalfield splitting parameter, WDq and of the interelectron repulsion between the delectrons themselves. One might suppose that while WDq measures the strength ofthe interaction between the metal d electrons and their ligand environment, theinterelectron repulsion is merely a property of the metal itself. That is not so,however, for the parameter B measures the d-d interactions in the metal within itsparticular environment. Interelectron repulsion energies are every bit as much aprobe of the molecular environment of a metal ion as are crystal-field energies. Forany given metal complex with a d2, d3, d1 or d8 configuration*, careful analysis ofexperimental transition energies yields values of both B and Dq. Such analyseshave been performed for scores, if not hundreds, of transition-metal spectra. So faras the B parameters are concerned, two general observations have emerged: a) Bvalues for metal complexes are smaller than the values B0 for the correspondingfree ions and b) B values may be placed in essentially fixed orders related to ligandor metal. First, observation a), often written as

β = Β/Β0< 1 (6.1)

and called the nephelauxetic effect, expresses the fact that the repulsions betweenthe d electrons in a complex are less than those in the corresponding free ion. Thisimplies that the average distance between the d electrons in a complex is largerthan the average for the corresponding free ion. Filling out observation b), it is found that for a series of complexes with acommon metal, the nephelauxetic effect increases in the order given in Eq. (6.2)

Transition Metal Chemistry. M. Gerloch, E. C. Constable

The ordering of ligands in Eq. (6.2) is about the same, regardless of the centralmetal. Analogously for a series of complexes with a common set of ligands, thenephelauxetic effect increases in the order

nephelauxetic effect increasing—>

and, again, the ordering of metals is roughly independent of the ligand set. In fact,it is possible to present the nephelauxetic effect, very roughly, as a simple multi-plicative function of independent metal and ligand parameters (Eq. 6.4).

(50-5)/5o = (1- j3) « h(ligands) x Jk(metal) (6.4)

Table 6-1. Some typical h and k values.

Ion k Ligand

Co(ii) 0.24 6 Br 2.3

Box 6-1 Examples: a) Using the values in Table 6-1, we find the nephelauxetic reduction for [NiF6]4- to be 0.8 x 0.12 = 0.096, that is, BIB0 = 0.904 or that B in this complex is reduced by about 10% relative to B0 for the Ni2+ ion. b) For [Co(NH3)6]3+, (1 -β ) = 1.4 x 0.35 = 0.49, that is, B in this complex is about half of that for the Co3+ ion.

Qualitatively, at least, there is a unifying theme and chemical correlation to be

found in these series. Namely, β values decrease with increasing reducing power ofthe ligands and/or increasing oxidizing power of the metal ions. These twostatements can be joined to yield the simple result:

B values decrease as negative charge is transferred from the ligands to the metal.

We can understand this powerful generalization directly from our view of thevalence shell in Werner-type complexes as laid out in Chapter 2. Recall that as anextreme limit for Werner-type species, we consider the metal contribution to thevalence shell for the first-row elements as 4s and 4p, with 3d orbitals excluded. So, 6.2 The Sp ectrο chemical Series 99

the bonds holding the complex together are very largely built from the 4s/4p metalorbitals together with appropriate ligand orbitals. As we pass from a free ion to acomplex, or as we traverse either of the nephelauxetic series (6.2) or (6.3), anincreasing amount of electron density is donated by the ligands into the 4s and/or4p metal orbitals. These metal orbitals are of the more penetrating type as theyhave subsidiary maxima fairly close to the metal nucleus (Fig. 2-2). Therefore, asmall but significant part of the donated ligand electron density enters thosepenetrating regions. That density lies between the metal nucleus and the bulk of the'innocent' 3d electrons. Consequently, the 3d electrons are more shielded from thenuclear charge in complexes than in the corresponding free ions, and more shieldedin complexes characterized by greater ligand —» metal electron donation than thosecharacterized by less. The greater shielding of the 3d orbitals results in their beingless well bound, and their radial distribution thus grows. The 3d orbitals grow morebulky and more diffuse, and the average distance between d electrons increases.Therefore, the average interelectron repulsion energy decreases, the 5/S0 valuesdecrease, and the nephelauxetic effect increases. The name nephelauxetic means 'cloud-expanding'. The explanation for it, whichwe have just reviewed, will be found elsewhere in the literature under the name'central-field covalency'. The magnitude of the nephelauxetic effect depends uponthe metal, its oxidation state, and the ligands bound to it as summarized in Eqs.(6.2) and (6.3) and Table 6-1. For hexaaquo complexes of various first-row metal(n)ions, for example, BIB0 ranges from 0.8 to 0.9; for many iodo or sulfur donorligands, one finds BfB0 values in the range 0.5 to 0.7; in certain low-spin cobalt(n)compounds, BIB0 values as low as 0.1-0.3 have been observed. However, becauseof certain artefacts in the way the B parameter is defined, none of these ne-phelauxetic effects should be viewed as implying more than modest expansions ofthe 3d electron clouds. Not the least of such considerations is the fact that theinterelectron repulsion energy varies inversely with respect to the electron - electronseparation so that we get a reciprocal relationship between B and the degree ofcloud expansion, rather than a linear one.

6.2 The Spectrochemical Series

The magnitude of Dq in any given complex is clearly a direct measure of the

interaction between the 'spectral' metal d electrons and their molecular environment.As for the nephelauxetic effect, values of Dq have been collated for a large numberof species and found to fit, very approximately, another multiplicative relationshipof metal and ligand functions (Eq. 6.5).

Dq -/(ligands) x g(metal) (6.5)

Again, ligands may be ordered according to the magnitude of Dq roughly

These orderings are called the spectrochemical series. At a purely empirical level,the collection o f / a n d g values (Eq. 6.5) in Table 6-2 is reasonably adequate topredict Aoct values for various metal-ligand combinations that did not define it. The-se values are also useful in connection with an empirical relationship known as thethe law of average environment. This law asserts that the splitting parameter Dq fora metal complex with a mixed set of ligands is given by the appropriately weightedaverage of the corresponding unmixed complexes. For example, from Table 6-2,4>ct for [NiF6]4- is 8010 cnr1 and for [Ni(H2O)6J2+ it is 8900 cm-1; the law ofaverage environment predicts that Aoct for [NiF4(H2O)2]2" is 8806 cm"1.

Box 6-2 This procedure is strictly invalid, of course, since the symmetry of a six-coordinate complex with dissimilar ligands cannot be exactly octahedral. In this case, further splitting of the d orbitals takes place which is not representable by a single splitting parameter like Aoct. However, if the departure from Oh symmetry is slight, so that spectral bands are broadened rather than split, the law of average environments retains utility.

Table 6-2. Some typical/and g values.

Ion g Ligand

Co(Ii) 9.3 6 Br 0.76

An explanation for the nephelauxetic series came readily to hand. Lets see howsuccessfully we can provide one for the spectrochemical series. First, note that Zioctvalues increase with decreasing size of the donor halides:

I < Br < Cl- < F-

and this seems reasonable in terms of the simple crystal-field model set out inSection 3.1 in which shorter bonds would indeed imply larger values of Aoct.However, the ability of the crystal-field model to rationalize the spectrochemicalseries stops right there. Thus, we might also expect orbital splitting energies to varyaccording to the charge on the ligands. We observe however, from Eq. (6.6) that thenegatively charged halogens produce less orbital splitting than neutral water orammonia ligands. Secondly, note that H2O, OH" and O2~ ligands define very similarZ\oct values. We then observe from Eq. (6.7) that z\oct increases with metal oxidationstate (Eq. 6.9).

M(II) « M(iii) < M(IV)

zloct increases-> (6.9)

This would not be expected simply on the basis of a crystal-field model, for the dorbitals will contract with increasing positive charge and hence interact less wellwith the ligand 'point charges'. The modest decreases in bond length as one traver-ses the series (Eq. 6.9) are unlikely to compensate for, let alone override, the effectsof such orbital contraction. Finally, to add to the confusion, we also note from Eq.(6.7) that A>ct values increase as we go down the periodic table (Eq. 6.10).

3d « 4d < 5d zloct increases—> (6.10)

This might be compatible with the electrostatic model in that the radial extensionsof 4d and 5d orbitals are greater than that of 3 J; but then the diffuseness of theseorbitals increases along the series in Eq. (6.10) and that would tend to decrease thezloct values. These and many similar observations made over the years all make it clear thatthe simple electrostatic basis of the pure crystal-field model utterly fails to provideeven a qualitative understanding of the spectrochemical series. This failure in noway casts doubt upon our successful interpretation of the nephelauxetic effect above.This is because crjsta/-field theory is incompatible with our views about bondingin the valence shell. Should the reader object to having been 'led up the gardenpath' for the past three chapters, he should remember that, notwithstanding the totalfailure of crystal-field theory to explain the magnitudes of the splitting parameters,their patterns and all properties flowing from them are accounted for withextraordinary success by this model. As we shall see, in replacing crystal-fieldtheory it would be stupid in the extreme to 'throw out the baby with the bathwater'.102 6 Ligand Fields, Bonding and the Valence Shell

6.3 Bonding in Octahedral Complexes

One might well wonder what crystal-field theory has to do with chemical bonding.After all, all interactions between metal and ligands are deemed repulsive: there isno mention of attractive binding forces. In this respect, crystal-field theory is hardlya chemical theory at all. To be fair, it was not developed to be one either. Thequalitative ideas of the approach, being essentially only dependent upon the dnconfiguration and molecular symmetry, are, however, quite compatible with bondingtheory, as we shortly describe. We shall also see how the quantitative aspects ofcrystal-field theory, as exemplified by the spectrochemical series, are illuminatedby a study of the bonding in transition-metal complexes. The approach we make isthrough a consideration of molecular orbitals in octahedral species.

6.3.1 Molecular Orbitals in Diatomic Molecules

Let us first briefly review the construction of molecular orbitals in simple diatomicmolecules, AB, using the linear combination of atomic orbitals (LCAO) scheme.The end product for the first long row of the periodic table is the well-knowndiagram in Fig. 6-1. We focus on two broad principles that are exploited in theconstruction of this diagram: one has to do with symmetry and overlap, the otherconcerns energies.

Figure 6-1. Schematic molecular

As to the first, we note the interaction of the s orbital of atom A with the sorbital of B, the pz with the ρZj and the px>y pair of A with the px>y pair of B. Inprinciple, of course, we could have considered the possibility of an interactionbetween, say, the s orbital on A with a px orbital on B as shown in Fig. 6-2. Thesketch shows that net overlap between these orbitals is zero and so no bonding orantibonding molecular orbitals are formed in this way. Now the labels s and px here

in-phase positiveoverlap

out-of-phase negative overlap

Figure 6-2. Zero net overlap between s (σ) and px (πχ) orbitals.

are symmetry labels for free atoms. Had we characterized these same orbitals withrespect to the cylindrical symmetry of the molecule to be formed, the s orbitalwould be labelled σ, and the px together with its partner py, would be labelled π.Then, using the rule that orbitals of different symmetry do not overlap, ourconclusion about the nonbonding interaction between these orbitals followsimmediately. While all this is well-known, and almost trivial in the present example,the classification of fragment orbitals according to the symmetry of the molecule tobe formed gains considerable utility in more complicated systems. Turning to the second point about energies, we recall how the stabilization of thebonding molecular orbital with respect to the lower of the two atomic orbitals, andthe destabilization of the antibonding one with respect to the higher lying atomicorbital, depend upon the magnitude of the relevant overlaps between the interactingatomic orbitals and upon their starting energies. Large bonding and antibondingenergy shifts are favoured by large overlap and/or good energy matching betweenthe relevant atomic orbitals. The smaller TT-TT* energy gap relative to the <7-σ*(from pz - pz overlap) in our example is expected in view of the less good 'sideways'overlap of two px orbitals relative to the 'head-on' overlap of two pz orbitals. We shall shortly draw on both of these symmetry and energy aspects of Fig. 6-1in the construction of molecular orbitals for the octahedron. First, however, let usextend the picture to molecules with more than two atoms.104 6 Ligand Fields, Bonding and the Valence Shell

6.3.2 Molecular Orbitals in Polyatomic Molecules

Figure 6-1 is a typical molecular orbital diagram in that molecular orbitals in themiddle are shown as arising from atomic orbitals on the left and right. The questionarises as to what an equivalent diagram might look like for, say, a triatomicmolecule. Should we construct one in three dimensions with atomic orbitals of Ashown on the left, those of B on the right, and those of C behind? It is possible. Butwhat do we do for a molecule with seven atoms as in an ML6 octahedral complex,for example? We could explore seven-dimensional diagrams! Well we don't do that.Instead, we consider the molecule as notionally broken into fragments which wethen consider two at a time. For a triatomic system ABC - linear or not - we couldproceed by one of three routes: 1) combine atomic orbitals of A and B to formfragment orbitals for the moiety A-B, and then combine these with atomic orbitalsof C to arrive finally at molecular orbitals for the complete ABC molecule, 2)combine B with C to get BC, followed by final combination with the atomic orbitalsof A or 3) combine A with C to form AC, and thence include B to get the samefinal result. A natural question at this stage is to ask whether one particular route isbetter than the others. But then, what is meant by 'better'? Presumably the bestroute is that which makes the most of any molecular symmetry and so yields therequired result in the shortest time. We illustrate this idea in the followingconstruction of molecular orbitals for the water molecule.

6.3.3 Molecular Orbitals for the Water Molecule*

The water molecule possesses two mirror planes of symmetry, as shown in Fig. 6-3.One mirror plane lies in the plane of the diagram through which the whole moleculereflects into itself across the plane. The other, through the oxygen nucleus in the yzplane of the figure, and shown by the dotted line, reflects Ha into Hb and vice versa.

ζ (into the page)

Figure 6-3. One mirror plane is the plane of the page, the other (—) is normal to the page.

We construct molecular orbitals for the complete H2O molecule, by first consideringcombinations of the symmetrically related hydrogen atomic orbitals, and then bycombining these with the oxygen atomic orbitals. We proceed by three steps: (A)combine the H Is orbitals and classify them according to their behaviour withrespect to the mirror planes discussed above, (B) classify the oxygen atomic orbitalswith respect to the same molecular symmetry, and (C) form H2O bonding andantibonding molecular orbitals by overlap of H-·-H and O orbitals of matchingsymmetry.

Step (A): Form combinations of H Is orbitals

yz mirror plane symmetry of combination

Type

Θ Is (i) symmetric with respect to reflection in

both yz and xz mirror planes

Is V J (ii) antisymmetric (changes sign) with respect

^—^ to yz mirror; symmetric with respect to plane of paper (xz)

Step (B): Classify O atomic orbitals

yz mirror plane symmetry of combination

Type

Is or

symmetric with respect to

both yz and xz mirror planes

(ii) antisymmetric with respect to yz ;

symmetric with respect to xz

(iii) symmetric with respect to yz\

but antisymmetric with respect to x.106 6 Ligand Fields, Bonding and the Valence Shell

Step (C): Combine like with like

bonding

symmetry V type (i)

antibonding

O bonding

symmetry type (i)

antibonding

O O OO bonding

symmetry type (ii)

OO antibonding 6.3 Bonding in Octahedral Complexes 107

The hydrogen orbitals do not form a combination of symmetry type (iii) and soleave the oxygen pz orbital as nonbonding. To complete the full molecular orbital diagram we should consider energies(determined in part by relevant overlap integrals) and the possibility of mixingbetween the molecular orbitals shown above as of like symmetry. However, weneed not bother with all that here. The purpose of this exercise has been to introducethe concept of group orbitals. In our example, we constructed two such grouporbitals - those in step (A) showing the constructive and destructive combinationsof H Is atomic orbitals. It is of no consequence that the magnitude of the overlapbetween these orbitals on well separated atoms is very small. We may still considera combination like [^1(H15) + 1/A2(H15)]/V2 to be a group orbital. Then, as we haveseen in step (C), these group orbitals are combined with symmetry matching oxygenatomic orbitals. Ignoring questions of relative energies and other quantitative matters,a final molecular orbital diagram (Fig. 6-4) may be constructed in a similar fashionto that shown for the diatomic case.

a.o. symmetry types group orbitals

for O of HoO m.o. for H-H

Figure 6-4. Schematic molecular orbital diagram for water.

We are now ready to apply the ideas in the preceding three sections to theconstruction of molecular orbitals in octahedral complexes.108 6 Ligand Fields, Bonding and the Valence Shell

6.3.4 The Molecular Orbital Diagram for Octahedral Complexes:

Local M-L σ Bonding

We begin by restricting consideration of the bonding in octahedral complexes to

interactions between the metal and any one ligand but involving only local σorbitals. By this we mean that we imagine each ligand to have only s or metal-directed ρ orbitals (or both) available for overlap with the metal. We label theavailable orbital on ligand i as σ,·. Further, and important, complications that arisewhen we include local ligand π orbitals into our scheme, are discussed in Section6.3.6. With this restriction to only local M-L σ bonding, we proceed as we did for thewater molecule above. Just as the two hydrogen atoms are there spatially related bythe molecular mirror symmetry, so here are the six ligands related by the molecularoctahedral symmetry. So, in step (A), we combine the ligand orbitals and classifythe resulting group orbitals according to the octahedral symmetry. Then, in step(B), we label the metal atomic orbitals according to that same symmetry, and finally,in step (C), we combine like with like. The results of pursuing steps (A), (B) and (C) are given in Table 6-3 and Figs. 6-5,6-6 and 6-7. Let us consider them in detail. In Fig. 6-5 we see how the combinationof ligand orbitals, (σι + (J2 + (T3 + σ4 + σ5 + σ 6 )/Λ/6, is symmetry matched to thecentral metal s orbital (the factor V6 is included simply to normalize this group

Table 6-3. Labelling of metal and ligand group orbitals in Oh symmetry.

Symmetry Metal orbital Ligand group orbital

dig s 1 / Λ/6 (σ, + G2 + σ3 + σ4 + σ5 + σ6)

eg dz2 1 / Λ/ϊϊ(2σ 3 + 2σ6 - σ} - σ2 - σ4 - σ5)

dx2_y2 1/2 ((Ji — 02 + (J4 — 05)

tiu Px 1/Λ/2(σ,-σ4)

py 1/Λ/2(σ2-σ5)

/7, 1/Λ/2(σ3-σ6)

orbital). The appropriate symmetry label for this combination in octahedral (Oh}parity is aig. Recall our use of these labels in Chapter 3: a means one-fold spatialdegeneracy, g means even (symmetric) under inversion through the centre ofsymmetry. In combining the metal s orbital with this ligand group orbital, weconstruct bonding (b) or antibonding (a) molecular orbitals for the complete ML6complex, depending upon whether the two orbitals (metal and ligand group) are inphase or out of phase: 6.3 Bonding in Octahedral Complexes 109

CF1+ CJ2 + 03 σ4 σ6) (6.11)

= a\x(s) - σ4 σ5 σ6)

where ^1, aj, ^1, b\ are all positive. (The reason why a\ Φ a\ and b\ Φ b\ derives fromthe inclusiqn of non-zero overlap integrals between metal and ligand group orbitals). The three-fold degenerate set of ρ orbitals are labelled flu (t for three-fold, u forodd under inversion through the centre of symmetry). As shown in Fig. 6-6, eachmetal ρ orbital matches symmetry with ligand group orbitals comprising just two

bonding α\% interaction antibonding a\g interaction

Figure 6-5. Interaction with the metal s orbital.

bonding antibonding

Figure 6-6. Bonding and antibonding ^ combinations are similar but with orbital densitiesalong the χ or y axes respectively.110 6 Ligand Fields, Bonding and the Valence Shell

out of the six available ligand σ functions. Bonding and antibonding tlu molecularorbitals for the octahedral molecule are then formed as

o(fii) = α2χ(Ρχ) (6.12)

together with equivalent (and degenerate) combinations between py with (O2-O5)

and between pz with (O3-O6). In Fig. 6-7, similar procedures are followed for the metal d orbitals. TheJX2 ^orbital-symmetry matches with the in-plane group combination (σι-ο 2 +ο 4 -ο 5 )/2to give the molecular orbitals described in Eq. (6-13).

y bonding e antibonding e£

bonding e^ antibonding e%

Figure 6-7. Interaction with the metal d x 2_ y 2 and dz2 orbitals.

6.3 Bonding in Octahedral Complexes 111

,- O2 (6.13) O4-O5)

2 The different shape of the dz orbital is matched by the ligand group orbital(2o3 + 2O6-Oi-O2-O4-O5)/^ίΪ2 , and we get the molecular orbitals in Eq. (6.14).

(6.14) *2 )-b'4(2o3 + 2σ 6 -σ 1 -σ 2 -σ4-σ 5 )

Finally, note that no combination of ligand σ orbitals interacts with members of

the metal t2g set. The vanishing overlap between any ligand ο orbital and, say, thed^ orbital is illustrated in Fig. 6-8. Overall, therefore, the metal t2g orbitals are non-bonding in this scheme. Recall how the 2pz orbital of oxygen is similarly nonbondingto the hydrogen orbitals in water.

Figure 6-8. Impossible symmetry matching of ligand σ orbitals with metal dxy Similar resultsapply for the xz and yz planes.

We now collect together the various parts illustrated in Figs. 6-5 - 6.7 by makingsome simple assumptions about the relative magnitudes of these metal-ligandinteractions. In this we refer to the arguments given in Section 2.2, namely, that thebonding interaction between the metal 4s orbital and the ligands will be greaterthan that for the metal 4p and that, because of radial compactness, the metal 3dorbitals will form the weakest interactions of all. Qualitatively, therefore, thecomplete molecular orbital diagram for a first-row octahedral complex with onlylocal M-L σ interactions is expected to like that in Fig. 6-9.112 6 Ligand Fields, Bonding and the Valence Shell

metal atomic ligand group

The final step is to house the electrons. Each ligand, acting as a σ donor, isconsidered to offer a lone pair of electrons. This is the case regardless of whetherthe ligand is formally negatively charged, like Cl", or neutral like NH3. Ultimately,these electrons are shared by the metal and ligand so that there is no implication,now or later, that the ligand 'donates away' or 'loses' two negative charges. In 6.3 Bonding in Octahedral Complexes 113

addition to these twelve electrons are η more, originating from the dn configurationof the metal, giving (12 + n) in all. Applying the Aufbau principle, we place twelveof these electrons in the lowest six bonding molecular orbitals, aig, tlu and eg, asindicated in the figure by crosses. The remaining η electrons then occupy thenonbonding t2g orbitals and the antibonding ef orbitals. We differentiate betweenthe first twelve and the remaining η electrons by crosses and arrows, not becausethese electrons are different in any way - for, of course, they are not - but toemphasize the connections between the present molecular orbital approach on theone hand, and the ideas of crystal-field theory on the other. Thus, the lowest sixfilled molecular orbitals provide an account of the binding, attractive forces betweenthe metal and the ligands. Above them lie first the t2g and then the (antibonding) e*orbitals, amongst which are distributed the same number of electrons as defined bythe metal dn configuration. In short, we may map this latter distribution onto thatdiscussed so fully earlier under the heading 'crystal-field theory'. Crystal-field theory accounts for the t2g-eg splitting, 4oct, in terms of the diffe-rential repulsion of the various electrons by ligands viewed as point charges. Withinthe molecular orbital scheme, on the other hand, that splitting is seen in terms ofthe antibonding energy of the e* molecular orbital (and of the t2g, as we shall see inthe next section). In turn, larger antibonding (repulsive) energies for the ef molecularorbitals are to be associated with larger bonding (attractive) energies for the egmolecular orbitals. Antibonding interactions are repulsive, as bonding ones areattractive. Immediately, therefore, we have one insight into the spectrochemical series bynoting that both bonding and antibonding energy shifts of eg and ef orbitals are

Box 6-3

L group orbitals

Weak M-L interactions lead to Strong M-L interactions lead to large

small stabilization of eg orbitals stabilization of eg orbitals and large and small destabilization of ef destabilization of ef114 6 Ligand Fields, Bonding and the Valence Shell

closely related to the donor ability of the ligands in the given complex. However,we need to do more work yet before we can construct any really successful accountof the spectrochemical series. We will return to this question shortly. Furthermore,we warn the reader that the immediately preceding discussion is somewhat naiveand must be re-addressed later (see Section 6.4).

6.3.5 Charge-Transfer Transitions

The 'd-d' bands involve electronic rearrangements within the t2g and eg (now ef)orbitals. The so-called 'charge-transfer' bands, on the other hand, involve transitionsbetween one or the other of these 'd' orbital subsets and other molecular orbitals inFig. 6-9. Some of these concern the promotion of an electron from the bonding tlMorbitals into the t2g or eg orbitals, others may concern electronic transitions from thet2g/eg orbitals into the t\% antibonding orbitals. We focus on these two types oftransition, rather than alg—> eg etc., as they alone amongst the many promotionspossible in Fig. 6-9 involve a parity change: u —» g or g —> u. Provided otherselection rules are satisfied, therefore, such transitions may be fully allowed. It isalso obvious from the qualitative ordering of molecular orbitals in the figure thatthese electronic transitions will occur at higher energies (larger promotion energies)than those of 'd-d' type transitions. We thus have an explanation for thegeneralization, exemplified in Section 2.1, that transition-metal spectra often showintense bands at higher energies than the weak 'd-d' bands. Recall here thediscussion about the perceived colours of the chloro- and bromocuprate ions inSection 2.1. The name 'charge-transfer' arises from the fact that these transitions take placebetween t2g or eg orbitals, which are largely of metal character (little mixed withany ligand orbitals), and members of the bonding or antibonding sets that possessvery much greater ligand character. Hence, these transitions involve a much greaterdisplacement of charge, one way or the other, between metal and ligand than do 'd-d' transitions. Finally, we must remember that just as a 'd-d' spectrum is not properly describedat the strong-field limit - that is, without recognition of interelectron repulsion andthe Coulomb operator - neither is a full account of the energies or number ofcharge-transfer bands provided by the present discussion. Just as a configurationt2g e™ gives rise to several terms, often with different spin, so also do excited lconfigurations like t2g~ eg ί λ } * . So we must expect the charge-transfer spectrum to beevery bit as complicated as the 'd-d'. While we do not pursue this complex matterfurther in this book, it is always well to keep in mind the fact that molecular orbitaldiagrams like that in Fig. 6-9 are but the beginning of any bonding picture.

6.3.6 Metal-Ligand π Bonding

Each local metal - ligand interaction in a complex might include a contribution fromπ bonding. We now remove the restriction of only local σ bonding adopted above, 6.3 Bonding in Octahedral Complexes 115

and consider the contribution of π bonding ligands to our molecular orbital scheme.Important differences arise between ligands as π donors or as π acceptors and westudy these two situations side-by-side. For concreteness' sake, we may envisagethe π orbitals on either type of ligand as ρ functions on the ligating atoms directednormal to the local M-L vector. Combinations of ligand π orbitals form only onegroup of importance that is symmetry matched to metal orbitals. These grouporbitals, of t2g symmetry in the octahedron, comprise a degenerate set of three andare shown in Fig. 6-10 together with the appropriate members of the t2g set of metald orbitals with which they overlap. We now incorporate the bonding and antibonding t2g molecular orbitals of Fig.6-10 into the energy diagram of Fig. 6-9. So as not to obscure the important issues

GXD bonding anti-bonding

'2k

Figure 6-10. Symmetry matching of metal t2g orbitals with ligand π functions. Similar diagramsmay be drawn in the xz and yz planes.

Box 6-4 Other group orbitals deriving from ligand π functions transform as f ] M , t\g, and t2u. The t2u group orbital can match the symmetry of some of the metal / orbitals and the t]g matches some metal g orbitals; we are concerned with neither of these here. The t[u ligand group orbital can overlap with the tlu set of metal ρ orbitals already used for σ bonding. Any such overlap is expected to be small, however, involving as it does 'sideways' overlap of metal ρ and ligand ρ orbitals on well separated centres. Accordingly, we also neglect this contribution to our bonding scheme, but here only for reasons of simplicity.116 6 Ligand Fields, Bonding and the Valence Shell

that arise, we draw incomplete diagrams, as in Fig. 6-11, in which only the egorbitals of the σ-bonding scheme are made explicit. Two diagrams are presented,one for each of the two cases of ligand η donors and η acceptors. In Fig. 6-11 a, werecognize that the π orbitals of ττ-donor ligands will be of relatively low energy(like the σ-donor orbitals) and be filled in the free ligands. Consequently, the bond-ing octahedral t2g molecular orbitals will also be filled. The crystal-field splittingparameter, zloct, is now identified with the energy gap t2*-e*. Recall how crystal-field theory considers all d orbitals as repelled by the ligands, but the eg more thanthe t2g subset because they point directly at the ligands. In the molecular orbital

(a) ligand π donors (b) ligand π acceptors

Figure 6-11. Metal - ligand π bonding in Oh symmetry. The boxed orbitals are filled withelectrons - nationally from the group of six ligands.

scheme of Fig. 6-11, the antibonding interaction with the eg orbitals is greater thanthat with the t2g, principally because σ overlap is greater than π. Finally, note thatZioct is less for a ligand which is both a σ- and ;r-donor than for a pure cr-donor.Greater σ bonding increases the energy of the ef orbitals and Z\oct, while greater πbonding increases the energy of the t2f orbitals and hence decreases Aoct. It is these 6.4 Ligand-Field Theory 117

opposing trends that we see as partly responsible for the lower position (smallerzloct) of the halogens in the spectrochemical series than that of the ττ-neutral ammonialigand, for example. In Fig. 6-lib we consider the case of ττ-acid (acceptor) ligands. Here, the ligandfunctions are empty high-energy ligand molecular orbitals we label TT*. We willlook at an example shortly. The stacking of the octahedral molecular orbitals nowtakes on the different ordering shown in Fig. 6-lib and, because the bonding t2gorbitals are empty (the ligands are π-accepting now), the crystal-field splitting, Zioct,is associated with the energy gap t2g-e*. This time, the greater the ττ-acidity of theligand set, the lower the energy of the bonding t2g orbitals and the greater Zioct. Theposition of the CO ligand at the higher end of the spectrochemical series is ascribedto their strong ^-accepting role. Overall, therefore, study of the bonding in octahedralcomplexes by molecular orbital methods predicts that Z\oct will be smaller for n-donor ligands and larger for ττ-acceptors, with ττ-neutral species inbetween. Broadly,that is what is found experimentally.

6.4 Ligand-Field Theory

The discussions of the past few sections are often cited to define ligand-field theoryas the application of molecular orbital theory to transition-metal complexes. It is nosuch thing. Certainly we can see how, from a first look at chemical bonding bymolecular orbital methods, one should not take a literal, that is pure electrostatic,view of the origins of crystal fields. Apart from those already raised in Section 6.2,there are abundant objections to the simplistic crystal-field view. Numericalcomputations of the magnitude of Zioct within a variety of electrostatic models haveyielded widely disparate estimates - some even of the wrong sign. As Van Vleckmade clear in 1935, the important conclusion to be drawn from schematic studieslike those described above is that the quantity Aoct is to be viewed as a parameter ofthe system which, inter alia, subsumes the consequences of covalent bonding. Mostimportantly, note too how such bonding could be dominated by non-J orbitals justas much as being a roughly equal property of all orbitals. Let us look a little more closely at this last point. Suppose, for the sake ofargument, that we take the extreme viewpoint mentioned in Section 2.2 that the 3dorbitals of a first-row transition metal (in higher oxidation states, remember) are socontracted that they effectively do not overlap with the ligand orbitals at all. Thiswould imply that the t2g-ef energy gap in Fig. 6-9 should be vanishingly small.Suppose further that the complex under consideration is one formed with neutral,rather than negatively charged, ligands. Any pure crystal-field (point-charge) splittingof the t2g and eg orbital subsets would similarly be expected to be very small. Evenunder these combined circumstances, however, we should still expect to observe asignificant t2g-eg energy splitting. That would arise from the Coulombic interactionof the d electrons with the non-spherical environment. For, don't forget, even if the3d orbitals overlap insignificantly with the ligands, the same is not true of the 4s118 6 Ligand Fields, Bonding and the Valence Shell

Box 6-5 By the way, recall the trend in Eq.(6.9) which was at odds with the crystal-field premise. Within the ligand-f'ield picture, both bonding electron density and (in the limit, nonbonding) J-electron density are increasingly drawn in towards the metal nucleus with increasing formal metal charge. These two electron densities are accordingly brought into closer mutual proximity and consequently, Dq values increase along this series.

(or 4p). That overlap, and bonding, is what holds the complex together and thebonding electron distribution which results is octahedrally deployed, not spherically.All in all, therefore, we would find the t2g and eg subsets of the 3d electronsenergetically differentiated by their differing proximities to the bonding electrondensity. In effect, the d electrons may be thought of as repelled by the bonds ratherthan by the ligands as point charges. Since the bonding electron density takes largevalues even in those regions reasonably close to the metal nucleus, the d-electron-bonding electron interaction is expected to be significant. Of course, a numericalcomputation of the t2g-eg splitting would be an extremely complex affair, not leastbecause it would be predicated on a full all-electron calculation of the bondingitself. Nevertheless, even without such a calculation it is apparent that the quantitywe call Aoct, though reflecting the underlying bonding in a given complex, must notbe naively thought of as simply - perhaps linearly - related to the t2g-eg splittingthat one might compute in a construction of a schematic diagram like Fig. 6-9. Notetoo that any differences between the schematic molecular orbital diagram and a fullall-electron calculation (the latter is, in any case, virtually impossible with currentcomputational facilities) are likely to be large with respect to the scale of the ligand-field energies. So, ligand-field theory is the name given to crystal-field theory that is freelyparameterized. The centrally important point is that ligand-field calculations,whether numerical or merely qualitative, explicitly or implicitly employ a ligand-field Hamiltonian, very much like the crystal-field Hamiltonian, operating upon abasis set of pure d orbitals. Instead of the crystal-field Hamiltonian (Eq. 6.15),

in which VCp takes a form describing the potential energy established by an array ofpoint charges, for example, we use the ligand-field Hamiltonian (Eq. 6.16),

XLF^Y^Vaj} +VLF (6.16)

in which VLF is an effective one-electron operator called the ligand-field potential

Note also that interelectron repulsions are no longer calculated with the explicitCoulomb operator, as in Eq. (6.15), but by an effective, two-electron operator U(i,j).This change is commensurate with the nephelauxetic effect, in which free-ion B0values are replaced with B values in the complex, in recognition of electron densitychanges brought about by covalency. 6.4 Ligand-Field Theory 119

Box 6-6 A simple example of an 'effective' operator with which the reader will be familiar is the use of Zeff elr as the effective nuclear potential experienced by an electron outside of a closed inner shell. Thus, we may compute the energies and wavefunctions for a 2s or 2p electron outside a Is2 shell, using the 'hydrogen-like' Hamiltonian,

but note that the value of Zeff is different for an outer 2s electron compared with a 2p electron. Well-known discussions of this difference centre upon the concept of variable shielding and orbital penetration, of course.

It is no part of our thesis in this book to get too technical. At the same time,however, it is surely unacceptable that a qualitative approach should avoid allmention of technicalities if that tactic results in a complete misunderstanding of thequite different standings of ligand-field theory on the one hand, and molecular-orbital theory on the other. Molecular orbital based discussions, like those in theimmediately preceding sections, provide insight into some of the trends in, andfactors affecting, ligand-field parameters. However, these two models do not maponto one another as computational procedures. Furthermore, the ligand-fieldapproach is closer to the end result so far as d electron properties are concerned,than are various conventional molecular orbital schema. Molecular orbital calculations may employ any convenient basis and, in many-electron applications, those bases will generally include within them somerecognition of all kinds of two-electron interactions. Metal s, ρ and d functions, forexample, will be treated in this regard on essentially equal footings. In ligand-fieldcalculations, on the other hand, all manipulations are made with respect to a pure dbasis, this being defined only with respect to the angular momentum property, beinggiven by / = 2, with no required statement about the radial part which mightotherwise define a free-ion 3d function. Furthermore, d-d interaction energies areexplicitly dealt with separately from all others (so we have separate parameters,like B for 'd-d' interelectron repulsions on the one hand, and Aoct for the ligand-field on the other). One is to realize that it is, a priori, strange and unexpected tofind that useful calculations of 'd-d' spectral transition energies and magneticproperties, amongst others, can be performed in this way. The d orbitals are explicitin such calculations while all others (metal s, /?; ligand functions) are left implicitwithin the effective operators of Eq. (6.16). Molecular orbital calculations, on theother hand, include all reasonable orbitals explicitly within a chosen basis set andthe operators are direct in the sense that they do not contain within them any otherbasis functions. Molecular orbital calculations are thus formally able to providequantitative accounts of all molecular properties ab initio while ligand-field theoryrelates only to J-electron properties, and then only in a parametric fashion. Fromthis remark, it might appear that the molecular orbital technique is clearly thesuperior. However, within the domain of transition-metal 'd-d' spectra andmagnetism, the fact is that, while ligand-field theory uniformly and consistentlyprovides a quantitative account of experiment together with much insight into the120 6 Ligand Fields, Bonding and the Valence Shell

too complex to compete at all in practice. It is equally evident that empiricalmolecular orbital methods are also ill-suited to this special range of properties. There is a parallel to ligand-field theory elsewhere in chemistry. The Hiickeltheory of organic π electron systems analogously focuses on just one subset ofelectrons. While various π orbitals are recognized explicitly, the roles of the G-bonding framework in such species are kept implicit, being 'folded into' the effectiveoperators of the model. The Hiickel approach (and here we mean the 'simple' Huckelmodel rather than various Extended Huckel approaches) is extraordinarily successfulas a parametric model of a particular set of electrons. The successes of Huckeltheory and ligand-field theory, strictly within their own domains, is owed to theway in which Nature keeps separate the relevant electronic sets. In ττ-electron theory,orthogonality is of the essence; in ligand-field theory, the (less perfect) decouplingof d electrons from all others derives from their relatively contracted nature, asdiscussed in Section 2.2. Were the d electrons less decoupled, i.e. more admixedwith the valence shell, ligand-field theory simply would not work. By 'work', wedon't just mean qualitatively, as established perhaps merely by symmetry, butquantitatively. It is unfortunate that the space and level of presentation prevent ourjustifying and exemplifying that ligand-field theory really does work at a quantitati-ve level: we ask the reader to be assured that it is so. Finally, on this question of the efficacy of ligand-field theory depending onNature's selection of a relatively isolated subset of electrons, consider what mightbe the limits to the domain of this approach. As will be discussed briefly in Chapter 10, the / orbitals in lanthanoide complexes are even more 'buried' beneath thevalence shell than are the d electrons in the main transition series. It will be of nosurprise, therefore, to learn that ligand-field studies on the spectra and magnetismof lanthanoide complexes, though technically rather more complex than for the dblock species, are entirely successful. On the other hand, suggestions which havebeen made in the literature from time-to-time that ligand-field techniques might beapplicable to /7-block compounds are ill-founded. There is no corresponding set of ρelectrons that is well isolated from other electrons in those systems. Similarly,ligand-field theory is not applicable to the charge-transfer spectra described inSection 6.3.5 for there we stepped outside of the d shell and included members ofthe valence shell itself. The above are clear-cut examples of the applicability ofligand-field theory; of the domain or 'regime' of the theory. Less clear areas alsocome to mind in which the approach may gradually begin to fail. These mightinclude, perhaps, the periphery of the d block, perhaps the third transition series.Paradoxically, and unfortunately, it is difficult to test this proposition since in thosesystems, charge-transfer spectra frequently obscure the 'd-d' bands whose analysismight provide the answer. To circumvent this, we might consider complexes in lowoxidation states. Indeed we have at last reached that long promised topic. 6.5 Synergic Back-Bonding 121

6.5 Synergic Back-Bonding

Just as the statement that 'such-and-such a compound is stable' is meaningless

unless one adds 'with respect to' something, so also is the definition of a ligand asa 'good donor'. A dramatic illustration of this idea is provided by carbon monoxide.A molecular orbital diagram for the free CO molecule is shown in Fig. 6-12. Wenote that the highest occupied molecular orbital (HOMO), 3σ, is a σ-bonding orbi-tal extending beyond the internuclear vector:

a.o. m.o. a.o.

for for for C CO O

Figure 6-12. Schematic molecular orbital diagram for carbon monoxide.

122 6 Ligand Fields, Bonding and the Valence Shell

The lobes of electron density outside the C-O vector thus offer σ-donor lone-paircharacter. Surprisingly, carbon monoxide does not form particularly stable complexeswith BF3 or with main group metals such as potassium or magnesium. Yet transition-metal complexes with carbon monoxide are known by the thousand. In all cases,the CO ligands are bound to the metal through the carbon atom and the complexesare called carbonyls. Furthermore, the metals occur most usually in low formaloxidation states. Dewar, Chatt and Duncanson have described a bonding scheme forthe metal-CO interaction that successfully accounts for the formation and propertiesof these transition-metal carbonyls. We see from Figure 6-12 that free carbon monoxide is bound in both σ and πmodes. As oxygen is more electronegative than carbon, resulting in the lowerenergies of the oxygen atomic orbitals than the carbon ones in the figure, the bondingTT molecular orbital, In, favours oxygen:

filled πone member of the bondingΙπιη.ο. in CO(the other member is normal to the page)

of the electroneutrality principle are then met by the CO group donating this chargeback to the metal via its now expanded cr-donor orbital:

δ-

Enhanced σ donor ability of the

^ towards the enhanced η accepting metal.

The effect is cyclic or synergic. The σ donation accumulates charge on the metalwhich donates back to the π* orbital of the carbonyl. An equilibrium is reached inwhich the (perhaps) neutral metal becomes bound to the neutral, 'poor σ donor'carbonyl by both strong crand π bonds. The α-donor function of the carbonyl groupis enhanced, or established, by the π acidity of this ligand, and by the enhanced σacidity of the metal brought about by its ability to rid itself of excess charge through'back-bonding'. The synergic back-bonding mechanism would fail if the metal werein too high an oxidation state for it would then lack the essential electron density todonate back to the carbonyl and so establish that ligand's cr-donor ability.

Box 6-7 Some prefer to introduce the back-bonding model by arguing first that CT donation from the carbonyl ligand causes too great an accumulation of negative charge on the metal so that the metal then tends to establish its electroneutrality by back-donation to the carbonyl π* orbitals. Since the whole process is synergic, it matters little at what point in the cycle one begins the description. However, the present, perhaps unusual, path was chosen so as to highlight the initial poor σ donor ability of the carbonyl ligand. Again, this is, no doubt, a matter of taste....

A similar account of the bonding in Zeise's salt, K[PtCl3(C2H4)], is offered by

the Dewar-Chatt-Duncanson model. In this complex, the ethene ligand bonds 'side-on' to the metal. The synergic back-bonding, shown in Fig. 6-13, involves donationfrom the filled π molecular orbital of the ethene together with back-bonding fromthe metal into the empty π* orbital of the ligand. This bonding mechanism is expected for complexes of electron-rich metals withligands offering both σ-donor and TT-acceptor functions. The underlying drivingforce for synergic back-bonding derives from the operation of the electroneutralityprinciple together with the existence of two (or more) discrete electronic pathwaysto satisfy it. The Dewar-Chatt-Duncanson model is widely accepted. However, itprovides us with a problem at this point. We have repeatedly emphasised the mini-mal roles of metal d orbitals in overlap with the ligands and yet the back-bondingmechanism can only succeed if the metal d orbitals overlap with the various ligandfunctions. It is curious that some argue that the (undoubted) validity of the Dewar-Chatt-Duncanson model demonstrates a universally active role for metal d functionsin metal-ligand bonding orbitals. There is, however, another way.124 6 Ligand Fields, Bonding and the Valence Shell

(a)

(b)

Figure 6-13. Synergic back-bonding in a platinum alkene complex. In (a), the interaction of a(filled) platinum 5d orbital with the TT* molecular orbital of the alkene is shown, whilst in (b),the interaction of a dsp hybrid orbital with the π molecular orbital of the alkene is shown. Notethat the two interactions result in electron density moving in opposite directions.

6.6 Valence Shells in High and Low Oxidation States

That way is to recognize that the nature of the valence shell is not constant butvaries throughout the transition-metal series as, indeed, do most chemical prioritiesthroughout the periodic table. The valence shell for Werner-type complexes,described in Section 2.2 in its limiting form, excludes the relatively tightly bound3d orbitals. The exclusion is not total, of course, because of the tail of the 3d radialwavefunction and as evidenced by the (small) 'violation' of Laporte's rule. Oneexpects the situation to be different in low oxidation state complexes like thecarbonyls. Here, the lesser formal effective nuclear charge results in all electrons -3d, 4s and 4p orbitals - being less well bound, but to differing degrees. From ourdiscussion of the electron configurations of transition-metal atoms in Chapter 1, forexample, we learned that the 4s electrons are often more strongly bound than the 3delectrons. This arises, we recall, because of the more penetrating character of the 4sorbitals. They themselves are thus partly exposed to a higher effective nuclear chargeand also their penetration serves to screen the 3d electrons more from the nucleus.Qualitatively, therefore, the differences between the radial forms of 3J, 4s and 4porbitals in higher and lower oxidation states are expected to follow the trendsillustrated in Fig. 6.14. We see that, in low oxidation state species, the radial extentof the 3d orbitals is much more like that of the 4s and 4p orbitals than it is inhigher oxidation state complexes. The valence shell now comprises all these orbitalsets. Now the d orbitals can overlap significantly with appropriate ligand functionsand no conflict with the back-bonding model is evident. Furthermore, as the dorbitals are now fully engaged in the bonding process and exposed to the envi- 6.6 Valence Shells in High and Low Oxidation States 125

ronment, one does not expect to find stable, open d shells. Unpaired electrons, forexample, within the d or valence shells should be particularly unstable with respectto any process that fills the shells. In short, we expect to observe an 18-electronrule [2 x (5 + 1 + 3) for the 3d + 4s + 4p subshells] governing electron counts, justas one sees the 8-electron rule within the first long series of the periodic table.Configurations with open shells and unpaired electrons in Werner-type complexesare stable and common (indeed, the norm) because the open d shells essentially lieinside the valence shell. As observed in the introductory survey of Chapter 1, thereis no tendency towards organic-like free-radial behaviour with Werner-typecomplexes having unpaired electrons. With very few exceptions, the same is nottrue in carbonyl-type chemistry of transition metals in low oxidation states. Overall,therefore, we argue that the change from higher to lower oxidation state chemistrysignals an important change of bond type in a way that the change from main groupto Werner-type transition-metal chemistry does not. This change in bond type, though discontinuous, is blurred. The passage fromone oxidation state to another is discontinuous in the sense that it is associated withdiscrete additions or removals of individual electrons. It is blurred, on the otherhand, because the electroneutrality principle will minimize any local chargeconcentrations. We saw one example of this in Section 6.1. The nephelauxetic effectarises because the radial distribution of the d shell expands as ligands donate nega-tive charge into the more penetrating regions of the 4s and/or 4p shells. Thenephelauxetic effect thus defines a spread of differential orbital expansion somewhatakin to the differences shown in Fig. 6.14. However, the range of radial variationfrom the nephelauxetic effect for any given formal oxidation state is, as commentedat the end of Section 6.1, generally expected to be less than that characterizingdifferent oxidation states. In Fig. 6-15, we provide a schematic indication of this'blurring'. Amongst the consequences to be expected from the change from Werner-typebehaviour to carbonyl, low oxidation state chemistry is a breakdown in the efficacy126 6 Ligand Fields, Bonding and the Valence Shell

range of nephelauxetic effect

3d radial , functions for functions for lower oxidation higher oxidation states states

Figure 6-15. Nephelauxetic expansions of the d orbitals are expected to be less than thoseresulting from integral changes in oxidation state.

of the ligand-field method. Calculations of spectral splittings between states which

are no longer well described by combinations of pure d functions cannot be expectedto succeed well at all. As discussed in Section 6.4, ligand-field theory works withina domain - the ligand-field regime, if you like - and lower oxidation state complexesare expected to fall increasingly outside of that domain. Unfortunately, it may notbe easy to demonstrate this expected failure experimentally. The greater orbitalmixing and larger d orbital splittings expected for the low oxidation state type ofvalence shell generally result in decreased energies of charge-transfer bands andgreater proximity of 'd-d' transitions to these charge-transfer bands. The result isthat spectral features that might be assigned as 'd-d' type are frequently obscuredby intense charge-transfer absorptions. A general paucity of apparent 'd-d'transitions thus precludes any thorough testing of the gradual breakdown of theligand-field method that is to be expected in low oxidation state transition-metalchemistry.

6.7 Electroneutrality and the Elasticity of the d Shell

The synergic, or interactive, nature of the back-bonding mechanism was strongly

emphasized in Section 6.5. Despite that emphasis, some find it natural to focusupon the change on the ligand: π back-donation by the metal enhances the σ basicityof the carbonyl group. However, one might equally observe that carbon monoxidebinds to transition metals because the metals act, in low oxidation states, asexceptionally good acceptors. Acceptors are atoms or molecules which readily absorbelectron density. Transition metals can do this either because they start out aspositively charged atoms (but not for carbonyls, of course, because these ligandsare intrinsically poor σ donors) or because they can divest themselves of the chargeby passing it onto other atoms (ligands) (see Chapter 9) or, indeed, back to theoriginal donor. There is even a third way in which transition metals can reduce 6.8 The Bonding Contributions of d Orbitals 127

charge concentration; namely, by the nephelauxetic effect. On receiving electron

density from one or more ligands, the expansion of the d shell, which, as we haveseen, may have little overlap with the ligand functions, spreads some of the metalelectron density out over a larger volume of space. As expressed in Chapter 1, theelectroneutrality principle asserts that atoms will acquire only small overall charge.However, the principle is more general than that, for it really means that chargewill be more evenly distributed. The enlargement of an atom, or of a subset of itselectron density, is an equally effective way of reducing charge density. The elasticityof approximately nonbonding d orbitals in transition metals confers upon the d-block metals an extraordinarily facile redox chemistry. Recall that an importantsubgroup of redox reactions in transition-metal chemistry is that involving stepwisegain or loss of electrons without bond rupture or overall geometry change. In part,this is to be laid at the door of elastic d orbitals. Of course, a great deal of redoxchemistry does involve bond rupture and ligand change, thus also characterizing thebroad spread of transition-metal chemistry, as it does elsewhere in the periodictable. Something of this is discussed in Chapter 9.

6.8 The Bonding Contributions of d Orbitals

We have emphasized the change of bond type that accompanies the growingparticipation of the metal d orbitals in the valence shell of transition-metalcomplexes on passing from high to low oxidation states. In order to make thispoint, we have perhaps overstressed the similarity between the Werner-typecomplexes in the d block and the chemistry of main group metals. We conclude thischapter, therefore, with the seeming volte face of asserting that, notwithstanding thesmall overlap of 3d orbitals with ligand orbitals in Werner-type compounds, theircontribution to overall bonding is by no means negligible. Once more, to make the point, let us take the extreme view that the 3d orbitalsoverlap negligibly with the ligand orbitals. Then, as described in Section 6.4, therepulsive or Coulombic interaction of the d electron density with that of the bondingelectrons results in their differentiation as monitored by ligand-field splittings. Aspointed out in Section 2.3 and explored in some detail in Chapter 7, that interactionimpinges on the bonds themselves. The physical approach of metal and ligands maybe frustrated to a greater or lesser extent by the d electron density. This hindrancewill depend upon the spatial distribution of the d electron density and, in turn, uponwhether the complex is in a 'high-spin' or 'low-spin' state, where that is appropriate.It is obvious then that the existence of the d electron density will generally have amarked effect upon net bond strengths and, on occasion, molecular geometry. Again,we shall see more of this in Chapter 7. In this sense, therefore, it would beappropriate to recognize that, even in higher oxidation state complexes, the d orbitalshave a significant role in bonding even where their overlap with the ligands may beminimal.128 6 Ligand Fields, Bonding and the Valence Shell

Furthermore, as discussed in Section 6.7, the ability of the elastic d orbitals to

function as electron 'sinks' contributes greatly to the rich variety of redox chemistrythat is so characteristic of the J-block elements. Here too, therefore, we recognizethe 'bonding' role of the d orbitals in Werner-type complexes as well as in carbonyl-type chemistry. In many respects, this is the kernel of this book. For years it has not been tooclear how one could consistently account for the wide variety of transition-metalchemistry in a way that does not conflict with the equally varied phenomena ofspectroscopy and magnetochemistry that are so well rationalized by ligand-fieldtheory. There is a tendency - psychologically quite natural, no doubt - for thoseinterested in synthetic and mainstream chemistry not to look too closely at theoryand physical properties, and, of course, vice versa. However, there has always beenthe need, surely, to build a logical synthesis of, or bridge between, these two aspectsof the same subject. We hope that our presentation in this book goes some waytowards providing that overview.

At the end of Chapter 2, we emphasized the interaction between the broadly non-bonding d electrons and all other electrons housed in the bonding orbitals of Wer-ner-type complexes. Most of the material we have covered since then has beenconcerned with the effects of the bonds upon the d electrons. Now we turn to theeffects of the d electrons upon the bonds. We shall see that, although the d orbitalsoverlap little with the ligand orbitals in Wernerian complexes, they do makesignificant contributions to what is collectively called the 'metal-ligand bonding'. In a nutshell, the d electrons repel the bonding electrons. They get in the way ofthe bonds and, to a greater or lesser degree, frustrate the attraction between metaland ligands. In essence, the proposed minimal overlap of d orbitals with the ligands,but significant repulsive interaction with the bonds, is equivalent to a focus uponthe two-electron operator rather than the one-electron operator; that is, uponrepulsions rather than overlap.

7.1 Bond Lengths in Octahedral Complexes

Consider the repulsive effects of the d electrons in a series of ML6 complexes as thed configuration of the central metal varies across the transition-metal series. All delectron density will repel the bonding electron density. The effects on the t2g electrondensity will be relatively small, however, as these orbitals largely lie inbetween thebonding regions. On the other hand, eg electron density directly frustrates thebonding. In Figure 7-1, we sketch the effects upon bond lengths predicted to arise fromthe repulsive role of the d shell in both high-spin and low-spin octahedral species.Of course, experimental bond lengths are also expected to decrease across the perioddue to the usual increase in Zeff that arises from the monotonic increase in nuclearcharge together with the imperfect self-shielding of non-core electrons. Figure 7-2presents typical variations in ionic radii as determined from experimental metal-ligand bond length measurements for both divalent and trivalent metals of the firsttransition period. The qualitative agreement between theory and experiment is evi-dent. Two further consequences of the steric activity of open d shells are also important.One, which might seem somewhat circular but does not, in fact, involve any 'doublecounting', is that longer bonds are accompanied by smaller ligand-fields, that is, by

Transition Metal Chemistry. M. Gerloch, E. C. Constable

Box 7-1 It is common to use plots of ionic radii, as in Fig. 7-2, for the transition metals as functions n of d configuration rather than the bond lengths. These are constructed by subtraction of appropriate ligand ionic radii from experimental bond lengths. As usual, one assumes an additivity between varying metal radii and a constant ligand radius. One might be concerned with two aspects of such a procedure: a) why use 'ionic' ligand radii rather than 'covalent', and b) might not the ligand 'radius' vary in response to the variable acidity of the metal across the series. We could avoid these imponderables by plotting typical bond lengths, noting that any variation in metal acceptor ability should vary with Zeff and the d configuration as in Fig. 7-1, but it is not usually possible to obtain a suitable series of complexes with similar ligands.

high spin

low spin

Figure 7-l.The repulsive effects of

the d shell on bond lengths. Small increases are expected with 3 4 5 6 7 10 occupancy of the t2g subset, large configuration ones with occupancy of the eg.

smaller Dq or Aoci values. The other is that variations in bond energies and theoverall thermodynamic stabilities of complexes will accompany these bond lengthvariations. We shall take up these themes in the next chapter.

7.2 Planar Coordination in d8 Complexes

Planar coordinated systems, you will recall from Chapter 1, formed a major groupof exceptions to the otherwise very successful geometry modelling of Kepert. Thatmodel explicitly neglected any steric role for the non bonding electrons, however.Let us now recognize and incorporate the steric activity of the d shell in J8 systems. First, consider an octahedral nickel(n) complex. The strong-field groundconfiguration is t^e\. The repulsive interaction between the filled t2g subshell andthe six octahedrally disposed bonds is cubically isotropic. That is to say, interactionsbetween the t2g electrons and the bonding electrons are the same with respect to Jt, yand ζ directions. The same is true of the interactions between the six ligands andthe exactly half-full eg subset. So, while the d electrons in octahedrally coordinatednickel(n) complexes will repel all bonding electrons, no differentiation betweenbonds is to be expected. Octahedral ds coordination, per se, is stable in this regard. Now consider a molecular stretching vibration that alternately elongates andcompresses axial (parallel to z, let's say) and equatorial bonds as outlined in Fig.7-3. Imagine an extreme vibration of this kind that eventually distorts an octahedralmolecule so as to gradually remove two trans ligands (again, let this direction be

Figure 7-3. Vibrational distortion leading to a planar complex.

taken as z). The ligand-field splitting diagram changes as illustrated in Fig. 7-4. Forligands regarded as point negative charges or as σ donors, the ligand-field along ζdecreases with respect to that along χ or y. A simple mnemonic is that "elongationalong ζ stabilizes orbitals with the letter ζ in them"; d^, is less repelled and morestabilized than J A 2 _ V 2 while dxz and dyz are more stable than dxy, although the splittingof the t2g orbital set is less than that of the eg set because the t2g orbitals are lessclosely directed towards the ligands. As shown in Fig. 7-4, a sufficiently large axialelongation of the octahedron, accompanied by a commensurate shortening of theequatorial bonds in response to the electroneutrality principle, raises the energy ofthe dj_y2 sufficiently that the electrons pair up in the d^ orbital, despite the penalty132 7 Steric Effects of Open d Shells

free ion octahedron elongated planar

octahedron

Figure 7-4. Schematic splitting of d orbitals in a planar environment. The ordering of thelowest four orbitals in unimportant here. The x2-y2 orbital lies much higher in energy than theother d orbitals, and the low-spin arrangement follows.

of pairing energy. In short, the molecule adopts a low-spin (xz, yz)4(xy)2(z2)2

configuration rather than the high-spin (xz, yz, xy)6(z2)\x2 -y2}1 configuration of thenear-octahedron. Now note that the doubly filled d^ orbital in the low-spinconfiguration offers two electrons' worth of repulsion along the ζ direction whilethe empty dj--y2 orbital provides no repulsion at all to the equatorial ligands. Inother words, the planar coordination, which we imagined as being achieved bydrawing out two trans ligands from a perfectly stable octahedron, is seen to bestable also with respect to d-electron repulsions. Furthermore, the d^ electron pairsituated above and below the coordination plane provides a strong disincentive tothe return of the axial ligands; that is to say, this nonbonding lone pair of electronstends to frustrate donor addition to, or adduct formation with, planar nickel(n)species. Also completing what at root is a cyclic, effectively synergic, process, wenote that the absence of J-electron repulsions from the dj_/ orbital encourages thecloser approach of the four ligands. This in turn allows these four ligands to satisfythe acidity of the metal atom as the six more distant ligands did in the octahedron.Altogether, therefore, we see that both octahedral and planar geometries are stablewith respect to the steric activity of the open d shell. 8 We also see how a planar geometry for J complexes can be preferred over atetrahedral one. With no regard to the steric role of the d shell, one expects, withKepert, to observe tetrahedral geometry for all four-coordinate complexes. On theother hand, should planar coordination be once achieved for the J8 configuration, itwill resist distortion towards the tetrahedron because of the repulsive effect of thelone pair normal to the plane. Kepert has observed that the placing of even one ortwo tenths of an electron charge between tetrahedrally disposed ligands on eachside of, and close to, the metal directly favours a switch to planar geometry. In 7.2 Planar Coordination in d8 Complexes 133

other words, four ligands plus two half lone pairs adopt an octahedral array. Inshort, Kepert's basic model needs to be supplemented by a recognition of the stericrole of an open d shell. If this is so, then how is it that his model works so well in other cases withoutthat addition? Well in many, though not all, cases, the additional effects of the d-electron repulsions are to modify bond lengths rather than bond angles. We discusssuch an example in the next section. Before doing so, however, there is more to sayabout planar coordination. We have argued that, once achieved, planar coordination in d* systems is stablewith respect to higher coordination number or tetrahedral distortion. The questionarises then about what circumstances favour planarity in the first place. In particular,we enquire about the occurrence of tetrahedral verses square planar stereochemistryfor d8 complexes. Why, for example, is the [Ni(CN)4]2" ion planar but [NiCl4]2"tetrahedral? First, note that there is a parallel relationship between high-spin tetrahedral J8and spin-paired planar J8, as compared with the octahedral and planar situationsjust described. Analogous to Fig. 7-4, we have Fig. 7-5. Do not be confused aboutthe reversed labelling of the xy and x2-y2 orbitals at the extremes of Fig. 7-4 and7-5 for the plane. The reversal is an artifact of the way we define the global axisframes for the tetrahedron and octahedron (see Figs. 3-2 and 3-6). Thus, onsquashing a tetrahedron to a square plane, we find the M-L bonds lying inbetweenthe χ and y axes while they lie along these axes for the situation depicted in Fig.7-4. Once again we see how the planar geometry is stabilized by removal of 'repul-sive' electrons from the dxy (ligand-directed) orbital. The achievement of planar

..'— xy -L ····"'"" u ....···"

·.!!__..···· xz,yz '"---....JL j_ ^~^~ Z

free ion tetrahedron flattened planar

tetrahedron

Figure 7-5. Compare with Fig. 7-4, but note the change of axis frame so that xy(oci)*2-.y2(tet).134 7 Steric Effects of Open d Shells

coordination again involves the pairing up of electrons and the contest between theligand-field promotion energy (E(xy) - E(x2-y2)) and the pairing energy. That contestis most likely to be resolved in favour of the planar, low-spin arrangement forligands higher in the spectrochemical series giving rise to greater ligand-fieldsplittings. Thus, we find planar [Ni(CN)4]2" but tetrahedral [NiCl4]2".

Box 7-2 In principle, we might expect to observe the low-spin J8 arrangement even in a distorted tetrahedral complex, provided that the associated ligand-field splittings are large enough. One such example is found in the complex [NiI2(Ph2PCH2CH2OCH2CH2OCH2CH2PPh2)]:

The change from high- to low-spin J8 configurations is necessarily discontinuous.

A given complex is either on one side of the divide or the other. We conclude thissection with a look at how the steric role of the d shell can affect angular geometrieswithin a series of just high-spin, nominally tetrahedral nickel(n) complexes. Before [Et4N]2[NiCl4], containing nearly tetrahedral [NiCl4]2" ions was firstsynthesized, it was thought that 'tetrahedral' geometry must be forced in four-coordinate nickel(n) species by including into the coordination shell such bulkygroups as triphenylphosphine. Indeed, the very first non-planar, four-coordinatenickel(n) complexes to be prepared were the bis-halo-bis(triphenylphosphine)nickel(n) molecules, NiX2(PPh3)2 X = Cl, Br, I). It was no surprise to find that noneof these molecules possessed very near tetrahedral symmetry. It was puzzling,however, to observe that their geometries deviate from tetrahedral towards planarcoordination increasingly along the series X = Cl, Br, I. In fact, the bis-iodo complexis planar coordinated. The increasing bulk of the halogens along this series mighthave led one to expect deviations from planarity that increase as iodine replacesbromine, or as bromine replaces chlorine. Clearly, the increased flattening of thetetrahedron on passing from the bis-chloro through to the bis-iodo complex occursin spite of, rather than because of, steric repulsion between the ligands. Recall, then, the positions of the halogens within the nephelauxetic series. Morenegative charge is donated to the central metal from the iodine ligands than fromthe bromine ligands, which in turn donate more than the chlorine ligands.Furthermore, it has been shown by modern ligand-field analysis that the extent ofthese charge donations is greater in the bis-phosphine complexes than in the tetra- 7.3 Trigonal Bipyramidal Coordination 135

halo complexes as a result of the π acidity of the phosphine ligands. Refer now to 8the splitting diagram for flattened tetrahedral J species in Fig. 7-5. While the fourt2 electrons in the pure tetrahedron are distributed equally amongst the three dorbitals, they are now arranged in favour of the dxz, dyz orbital pair. Conversely,there is less d electron density in the dxy orbital - the %' orbital most nearly directedat the ligands in the flattened tetrahedral coordination. Overall, the tendency of atetrahedron to distort towards a plane will be greater the more important the repul-sive interactions between the d electrons and the bond orbitals become. Thoserepulsions are expected to increase as the d orbitals expand in response to theincreasing nephelauxetic effect along the series Cl < Br < I. In short, we argue thatit is the d-electron- bonding electron repulsions that determine the relative angulargeometries of these bis-halo-bis-phosphine nickel(n) complexes.

7.3 Trigonal Bipyramidal Coordination

The steric activity of open d shells is well illustrated by the ligand fields and bondlengths in five-coordinate complexes with formal trigonal bipyramidal geometry.Consider the series of complexes [Mn(Me6tren)Br]+, where Me6tren is tris((dimethyl-amino)ethyl)amine. The Me6tren ligand is a tripodal tetraamine donor and thecoordination geometry of this series of complexes is shown in Fig. 7-6. Metal-

Co(ii) Ni(ii) Cu(ii) Zn(n)

M-N(ax)/ A 2.15 2.10 2.07 2.19

M-N(eqp/ A 2.08 2.13 .2.14 2.11M-Br/ A 2.43 2.47 2.39 2.45

xy, x2-y2

xz, yz

Consider first what pattern would be expected for the zinc(n) complex. Here, thed shell is full (J10) and therefore offers equal repulsion in all directions. As such,we should be able to make a prediction based solely upon bond-pair repulsions asin VSEPR theory. We predict that the axial Zn-N bond should be longer than theequatorial Zn-N bonds because it suffers three bond-bond repulsions at 90° whileeach equatorial bond suffers only two such repulsions at 90°. The experimentalbond lengths in Table 7-1 support this view. Now consider the cobalt(n) complex.In the three-fold symmetry of the trigonal bipyramid, the d orbitals split up asshown in Table 7-1. The d^ orbital is highest in energy because it points directly atthe ligands, the degenerate dxy, d^_y2 pair lies in the plane of the equatorial ligandsand the electrons in these two orbitals are the next most repelled, and the degeneratedxz, dyz pair is directed inbetween the ligands and is the most stable. All complexesin this series are high-spin. Their strong-field configurations are indicated in Table7-1. In each case, the dxz, dyz orbital subshell is full and, for the cobalt complex,there is one electron in each of the remaining three d orbitals (in effect, one in theJZ2 orbital directed mostly along ζ and two in the dxy, dxzi _/ pair involved, betweenthem, with the χ and y directions). The repulsions offered towards the axial andequatorial ligands by this electronic arrangement are thus essentially equal. Wetherefore expect to find the same relationship between axial and equatorial bondlengths in this cobalt(n) complex as in the zinc(n). The bond lengths in Table 7-1support this. Although their absolute values differ between the d1 and J10 complexes,of course, because of the overall trend in Zeff to increase across any period, we dofind longer axial and shorter equatorial Co-N bonds as predicted. Turning now to 9the d copper(n) complex, we note that the dxy, dx^ orbital pair is full while the d^orbital still houses only one electron. The d-orbital to ligand orbital repulsions inthe equatorial plane are thus much greater than along the three-fold axis and thebond length pattern is reversed. What is actually observed is a marked shortening ofthe axial bond. This results from the combined effects of the change in steric activity 7.4 The Jahn-Teller Effect 137

of the open d shell and of the increased effective nuclear charge on replacing cobaltby copper. Copper is the stronger acceptor and the drive towards electroneutrality issatisfied most easily by a close approach of the axial ligands since they are leasthindered by the d-electron distribution. These bond length variations are complemented by analogous changes in thestrengths of the local ligand fields associated with the various ligands. Local ligand-field strengths are represented and monitored in a version of the ligand-field modelcalled cellular ligand-field theory. While space and level do not allow any fulldiscussion of this powerful modern approach, some idea of its basis is presented inBox 7-3 for those with an interest.

7.4 The Jahn-Teller Effect

A somewhat abstruse group-theoretical (symmetry based) theorem was published in

the late 1930's by Jahn and Teller. It is in effect that "For non-linear molecules, anuclear configuration which begets an orbitally degenerate occupied state is unstablewith respect to one without such orbital degeneracy." In the 1950's, Orgel exploitedthis theorem to rationalize anomalous geometrical features of copper(n) andchromium(n) compounds. Empirically, it is found that formally octahedral ortetrahedral d9 complexes are highly distorted and, to a slightly lesser extent, thesame is true of high-spin octahedral d4 species also. Typical of these distortions isthat while equatorial metal - oxygen bond lengths, for example, in 'octahedral'copper complexes are about 2.0 A, axial bonds for two trans ligations take valuesanywhere between 2.3 and 2.9 A. These are large effects and are not observed incomplexes of metal ions with other dn configurations, unless caused by apparentligand constraints (chelation, for example). The explanation of this effect - generallyknown as the Jahn-Teller effect - exploits the Jahn-Teller theorem. In octahedral symmetry, the ground term of the d9 configuration is 2E8, asdiscussed in Chapter 3. This is an orbitally degenerate state and hence subject tosome nuclear distortion that removes the degeneracy. The situation is represented inFig. 7-7 where we investigate the effects of a tetragonal molecular distortion. Sincethe Jahn-Teller theorem does not determine the type of distortion that must occur,we look at the tetragonal one simply because most Octahedral' copper(n) distortionsroughly conform to this. In Fig. 7-7, we see that the e\ configuration of the regularoctahedron is degenerate (2£g) because the hole may be sited in the d? or dx2_y2orbitals with equal probability. Tetragonal elongation or compression of the octa-hedron removes the degeneracy of the eg orbital pair to leave 2A lg or 2Blg -corresponding to the hole being housed in the d^ or <i2_y2 orbital respectively -ground terms which are, of course, nondegenerate. The driving force for thedistortion is, once again, the steric activity of the open d shell. In high-spin species (to which the theorem is not restricted), perusal of theappropriate configurations for octahedral complexes across the transition period(see Fig. 5-1) shows that d4 and d9 configurations are candidates for the Jahn-Teller138 7 Steric Effects of Open d Shells

Box 7-3 Consider the local interactions between d orbitals, referred to a local frame, and various bond orbitals. In Fig. A, we represent such interactions for orbitals characterizing local σ

d72

M-L

M-L d72 ^dn σ bond orbital

Figure A

symmetry. The da orbital (dz2) suffers a shift in energy that we label ea (we use the lower- case e for the energy shift of an orbital.). The equivalent situation for a π interaction in the local xz plane is shown in Fig. B and defines the local ligand-field parameter em. An analogous

'\ M-L η bond orbital

Figure B

interaction in the yz plane defines e^ These various parameters relate directly to the nature of any σ, Jtx or jty bonding within the local region of space. Thus, for a metal-pyridine ligation, for example, one expects that bonding orbitals between metal (4s/4p) orbitals and ligand orbitals will exist for σ and n± (where J. means perpendicular to the pyridine plane) interactions but not for π// Detailed analysis confirms this sort of prediction. The (global) ligand field for the complex as a whole is constructed by appropriate additions of such local, or cellular, ligand fields. This powerful approach provides a means by which analysis of global ligand-field phenomena - spectral transition energies, intensities, optical activity as well as various magnetic properties - may probe the underlying bonding in a complex. 7.4 The Jahn-Teller Effect 139

xy

octahedron elongated compressed

Box 7-4 A tetragonal distortion is one that maintains the four-fold symmetry of the octahedron. Here, we consider two equal trans metal- ligand bond lengths being different from the other (equal) four. A trigonal distortion maintains the three-fold symmetry from the octahedron. This might be effected by six equal bond lengths with unequal interbond angles:

elongation compression

tettagonal distortions

a<90° a>90° β<90°

trigonal distortions140 7 Steric Effects of Open d Shells

effect because of orbital degeneracy in the eg configurations and J1, d2, d6, and d1configurations because of orbital degeneracy in their t2g shells. However, Jahn-Teller effects arising out of t2g shells that are incompletely (but not exactly half-)filled are much smaller in magnitude than those associated with e\ or e\configurations. This is because the t2g orbitals interact less with the bonds than dothe eg orbitals, being directed inbetween them. Once again, in evidence of theinteractive nature of d electrons in the ligand environment, the splitting of the t2gorbitals due to Jahn-Teller distortion is small and the distortion due to the unevenlyfilled t2g subshell is also small. We comment further on the difference betweenunevenly filled t2g and eg shells shortly. For octahedra, only d4 and d9 configurationsare expected - and observed - to suffer significant Jahn-Teller distortions. For tetrahedra, orbital degeneracies in the higher-lying t2 orbital subset shouldgive rise to larger distortions than in the e set. On this basis, one expects distortedtetrahedra for J3, d4, d8 and d9 configurations. Tetrahedral d3 and d4 complexes arevery rare. Tetrahedral d% complexes are reasonably common but occur often withreasonably regular geometries. It is likely that the lack of any significant Jahn-Teller distortion here is due to the strong-field limit being a poor description of themany-electron ground state in these systems. Tetrahedral copper(n), J9, complexes,on the other hand, show large distortions, most frequently in the form of a flatteningtowards square planar geometry (see Box 7-5). Returning to the octahedral species, there is one curious feature to the types ofdistortion observed in practice. The scheme in Fig. 7-8 shows how the distortedoctahedron acquires a stability over the regular octahedron in d4 and d9 systemsequal to one half of the splitting of the eg orbital pair. This arises as follows. Tomaintain the same mean ligand-field strength - in effect, to satisfy theelectroneutrality principle to the same extent - a lengthening of two trans bondlengths is accompanied by a (lesser) shortening of the four equatorial bonds, or viceversa: this results in a barycentre splitting of the eg orbitals as shown in Fig. 7-7. Ifwe label that splitting as 2<5 with respect to the energy of the octahedral eg orbitals,

Box 7-5

InCs 2 [CuCl 4 La= 130°

InCs2[CuBr4], a- 140C 7Λ The Jahn-Teller Effect 141

the dz2 is stabilized by <5 upon elongation, for example, while the J^ / is destabilizedby the same amount. The lower energy orbital houses two electrons (stabilization2<5 ) while the higher one houses just one electron (destabilization 5), giving anoverall stabilization of δ which is one half of the eg orbital splitting due to distortion.No stabilization is associated with the splitting of the t2g orbital set here becausethis set is full in d9 or exactly half-filled in d4. Since the foregoing argument issymmetrical with respect to elongation or compression of the octahedron, the labelson the two members of the eg set are irrelevant. We might therefore expect elongatedoctahedra to occur roughly as often as compressed ones. Empirically, however, verymany more elongated chromium(n) complexes are observed than compressed ones,and no compressed copper(n) complexes are observed at all (ignoring any with thatgeometry imposed by chelate ring strain and the like). Clearly, our explanation isincomplete. Crystal-field theory accounts for J-orbital energy shifts in terms of the differenti-al repulsive effects of negative point charges: the various d orbitals are raised inenergy. Ligand-field theory also refers to a raising of J-orbital energies, at leastwhen the bond orbitals with which they interact are lower in energy than the dorbitals: such is the case for ligands acting in σ- or π-donor modes. Occasionally,d-orbital energy shifts can be to lower energies, however. One case with which weare familiar is when a bonding orbital is higher in energy than the d orbital, as forligations in the ^-acceptor mode (see Section 6.3.6). Another case, not discussed sofar, is when a d orbital interacts significantly with another suitable orbital of higherenergy. Here we consider the antibonding 0lg orbital of predominantly 4s parentage(see Fig. 7-8). In strict octahedral symmetry, no proximate bonding or antibonding orbital arisingfrom outside the d shell has the same symmetry as the members of the d shell. Inthe tetragonal symmetry of the present distorted octahedra, however, this is nolonger true. The J-orbital symmetry labels in the tetragonal (D4/, point symmetrygroup) environment are: dj- (aig); dx^ (/?ig); dxy(b2g)\ dxzdyz(eg). The predominantly4s antibonding orbital, labelled alg in octahedral symmetry, is still of alg symmetryin the tetragonal environment. There arises, therefore, the possibility of interactionbetween, in effect, the metal 4s orbital and the dj- orbital, but only the dj- orbital.As the energy of the 4s orbital is higher than that of the d^- , the interaction is suchas to stabilize the dzi orbital further. This is true for both elongated and compressedoctahedral geometries as shown in Fig. 7-8. We can expect the extra stabilization,<5/ of the dz2 orbital to be roughly similar for either sense of distortion. However, inthe elongated octahedron, two electrons occupy the d^ but only one in thecompressed geometry. This provides the asymmetry we seek. The energy gapbetween the various d functions and the antibonding ^1* is expected to be muchless at the right end of a transition period than the left because of the morepenetrating character of the 4s metal orbital together with the increased effectivenuclear charge at the right end. So this extra stabilization, due to d-s interactions,that favours elongated geometries over compressed is expected to be more significantfor copper(n) complexes than for chromium(n). As noted above, both geometry 4types are possibly observed experimentally for the d system, but only the elongated 9one for d ·142 7 Steric Effects of Open d Shells

The driving force for Jahn-Teller distortions in transition-metal complexes is the

open d shell. It is likely that explanations for them along the lines given abovewould have come about even if the theorem of Jahn and Teller had not beendiscovered. We make this remark not to denigrate that powerful piece of work, butas an attempt to defuse any mystery that might otherwise attach to Orgel'sapplication of that group-theoretical construction. The smaller magnitudes of distortions associated with unevenly filled t2g subshellsin octahedral complexes relative to those deriving from unevenly filled eg subshellsinvites a little thought. From Fig. 7-7 we see how a distorted geometry for an openeg shell is more stable than an imdistorted one. The splitting, 25, is a measure ofthat relative stability. Suppose for the moment that one were to focus upon thatsplitting energy as an absolute goal that is to be achieved in various situations. Thatis to say, suppose the measure of the relative stability of the distorted over theundistorted geometry is just given by the low-symmetry field splitting. If so, wemight expect a larger physical distortion for open t2g shells than for open eg shells.That is because, for a given distortion, the splitting of the eg orbitals which pointdirectly at the ligands is greater than that of the t2g orbitals directed inbetween. Thispredicts a result which is contrary to empirical fact. If, on the other hand, we 7A The Jahn-Teller Effect 143

recognize the driving force for the distortion as the steric activity of the open dshell, we see immediately that the repulsive hindrance of t2g electron density is lessthan that of the ligand-directed eg density. An octahedral molecule with an open egshell will distort more than one with an open t2g shell. In consequence, certainorbital splittings occur and confirm, as it were, the distortion in the manner of Fig.7-7. However, there is no simple or direct way in which we can estimate themagnitudes of these orbital splittings and their associated stabilizing effects a priori. The Jahn-Teller effect is pervasive. We have described its manifestation inelectronic ground states leading to static distortion: the so-called 'static Jahn-Tellereffect'. When the tendency to distort involves smaller energies that are comparablewith either spin-orbit coupling or vibrational energies, static distortions may not beobserved. Instead, strong coupling of electronic and nuclear motions may result andgive rise to the 'dynamic Jahn-Teller effect". Unfortunately we cannot pursue thismatter here. We further note that 'unexpected' splittings are sometimes observed in'd-d' spectral bands. These have been ascribed to Jahn-Teller effects in excitedstates.

Box 7-6 Example: The spectrum of [Ti(H2O)6J3+ ions, whose ground state geometry is nearly perfectly octahedral, is characterized by a large splitting.

Only one band maximum is expected, of course, corresponding to the 2T2g-^ 2Eg transition. The splitting of the asymmetric peak is ascribed to a Jahn-Teller splitting of the excited state which latter involves the open eg configuration e\

Finally, we should recognize that there can also be second-order Jahn-Teller

effects. Above, we address orbital degeneracies within the ground state. However,since spin-orbit coupling, for example, can admix orbitally degenerate excited statesinto an orbitally non-degenerate ground state, Jahn-Teller effects can be observed insecond-order. Once again, we do no more than mention these matters here.144 7 Steric Effects of Open d Shells

Suggestions for further reading

- The scope of Kepert's model is, herein, made plain.2. J.S. Griffith, Theory of Transition Metal Ions, Cambridge University Press, Cambridge, 1961.3. CJ. Ballhausen, Molecular Electronic Structures of Transition Metal Complexes, McGraw-Hill, New York, 1979. - This last reference is for those who would like to see the Jahn-Teller theorem at a technical level.8 Complex Stability and Energetics

8.1 The Thermodynamic Stability of Complexes

The thermodynamic stability of coordination compounds is relatively easy to

determine, and provides us with a valuable pool of data from which we may assessthe importance of ligand-field and other effects upon the overall properties oftransition-metal compounds. The bulk of this chapter will be concerned with thethermodynamic stability of transition-metal compounds, but we will briefly considerkinetic factors at the close. The stability of a complex is conveniently expressed in terms of the thermo-dynamic stepwise stability constant K as defined in Eq. (8.1).

ML(n_1} + L=^ MLn /8 })

K= [MLn] /([ML^1)][L])

We should note at this point, that the above reaction implicitly refers to aqueoussolutions, and that, for convenience, we have explicitly excluded free andcoordinated solvent molecules. Strictly, the above relationships should be written asin Eq. (8.2).

M(H20)(7_Ai)L(,_1) + L^ M(H20)6_nLn + H2O (8 2)

K = [M(H2O)6^Ln][H2O] / ([M(H20)(7_n)L(n_1}][L])

For obvious reasons, we tend to use the simpler form, although we will discusssome of the limitations shortly. We may also consider overall stability constants, βη(Eq. 8.3).

M + nL=^ MLn /g βχ βη= [MLn] /[M][Lf

There is an obvious relationship between Kn and /Jn as expressed in Eq. (8.4).

^ (8.4) ο Transition-metal complexes span an enormous range of stabilities. One of theprincipal aims of this chapter is to attempt to understand some of the factors whichcontrol these, and to determine the importance of ligand-field effects. Very extensi-ve compilations of stability constants are available.

Transition Metal Chemistry. M. Gerloch, E. C. Constable

8.2 The Chelate Effect and Polydentate Ligands

Ligands containing more than one donor atom which can bond to a metal centre aretermed polydentate or multidentate. Such ligands are extremely important and haveplayed crucial roles in the development of coordination chemistry. Well knownexamples include 1,2-diaminoethane (ethylenediamine, en)', 2,4-pentanedionate(acetylacetonate, acac~)\ 2,2'-bipyridine (bpy)\ and l,2-diaminoethane-N,N,N\N'-tetraacetate (edta4') (Fig. 8-1).

/ \ H2N NH2

en

O2C N N CO2-

"O2C CO2~~

bpy edta'

Figure 8.1. Some common polydentate ligands.

When two or more donor atoms from the same ligand are coordinated to a singlemetal centre, the ligand is said to be chelating. It is a general observation thatchelated complexes of polydentate ligands are always more thermodynamicallystable than those of the same metal with an equivalent number of comparablemonodentate ligands. That is to say, the equilibrium

MLn + (LL)n/2^ M(LL)n/2 + nL

(L = monodentate ligand, LL = didentate ligand)

lies to the right. This is exemplified in the data for the Ni2+/ en/ NH3 systempresented in Table 8-1. 8.2 The Che late Effect and Poly dentate Ligands 147

Table 8-1. Stability constants for some nickel(n) complexes of ammonia and 1,2-diaminoethane.

M2+ AG/kJ mol- 1 AH/kJ mol-1 TAS (298K)/kJ mol-1

Mn -15.9 -11.7 4.2

M(H2O)62+ + en^± M(Cn)(H2O)42+ + 2H2O

In each case, both the entropy and enthalpy terms favour the formation of thechelated complex, regardless of the J-electron configuration. Note, however, thatoutside the d block, i.e. with alkaline earths and other main group metals, it is oftenfound that the entropy term is dominant.148 8 Complex Stability and Energetics

[Ca(H2O)6J2+ + edta4-^ [Ca(edta)]2- + 6H2O

AG/kJ mol-1 AHIk] moH TAS (298K)/kJ mol"1

-62.0 -26.0 36.0

In some cases, the enthalpy term may actually oppose the formation of thechelated complex, although the entropy term outweighs it to give an overallfavourable free energy term. In general, this situation is the exception rather thanthe rule.

[Co(H2O)6I2+ + [P3O10]5- [Co(P3O10)]3- + 6H2O

AG/kJ mol-1 ΔΗ/kJ mol"1 TAS (298K)/kJ

-45.0 18.8

8.2.2 Contributions to the Chelate Effect - The Enthalpy

When two ligand donor atoms are brought into proximity upon bonding to a metalion, an electrostatic repulsion between the negative charges or dipoles is experienced.In the case of two monodentate ligands, this repulsion increases as the ligands arebrought together, whereas in the case of a didentate ligand it is already 'built in'(Fig. 8-2).

H2N NH2 ' H2N NH2

\ / M

H H3N NH3 · > 3N NH3 MFigure 8-2. Schematic representation for the formation of a complex. In the case of themonodentate ligands, there is a greater unfavourable nitrogen-nitrogen repulsion involved inbringing the ligands together.

We must also consider the changes in solvation of the ligands which occur uponcoordination. If we consider an amine in water, we would anticipate stronghydrogen-bonding. If we compare 1,2-diaminoethane with ammonia, we wouldexpect the latter to be more highly solvated. This corresponds to a more unfavourableenthalpy associated with the desolvation. 8.3 Ligand-Field Stabilization Energies 149

8.2.3 Contributions to the Chelate Effect - The Entropy

The simplest way of thinking about the entropic contribution is to consider the'half-way' stage in the formation of a chelate complex (Fig. 8-3).

Figure 8-3. The final ring-closure step in the formation of a chelate.

In forming the chelate complex, there is a high probability of the second donoratom Y forming a bond to the metal whereas, with monodentate ligands, theprobability is much lower. In other words, once the first M-L bond is formed, thesecond donor atom is held close to the position required for the formation of thesecond bond. In more mathematical language, the favourable entropy term is associated withthe release of a large number of monodentate ligands upon the formation of thechelate.

[Cu(NH3)2(H2O)2]2+ + en=^ [Cu(NH3)2(en)]2+ + 2H2O

2 molecules 3 molecules

AG/kJ moL1 ΔΗ/kJ mol-1 TAS (298K)/kJ

-15.5 -8.0 8.64

There is also an entropy term associated with the desolvation of the ligands. This ismuch more difficult to assess, and may make for either favourable or unfavourablecontributions to the overall entropy changes. We now consider what ligand-field theory may contribute to an understanding ofthe variation in stabilities of transition-metal complexes as a function of the dconfiguration.

8.3 Ligand-Field Stabilization Energies

Recall the splitting of the d orbitals in octahedral environments. The energies of thet2g and eg subsets are shown in Fig. 8-4 with respect to their mean energy. We haveused the conventional 'barycentre' formalism. In effect, we express the energy ofan electron in the t2g or eg orbitals with respect to the total energy possessed by aset of five electrons equally distributed amongst the five d functions. Alternatively,we say that our reference energy is that of a d electron within the equivalentspherical mean field.150 8 Complex Stability and Energetics

+0.6AOC,

-0.4AOC1

Figure 8-4. Splitting of the d orbitals in an octahedral ligand field.

In Fig. 8-5, we illustrate the orbital occupancies in the strong-field limit for dl to 9d configurations in high- and low-spin arrangements. The J-orbital energies weassociate with each t2geg configuration are computed with respect to our barycentreorigin. For example, for the high-spin d4 configuration in octahedral symmetry,t2ge], we add up the energies of the d electrons as (-4 Dq) + (-4 Dq) + (-4 Dq) +(+6 Dq) to get -6 Dq (or -0.6 Aoct). We call these configuration energies Ligand-

IJ in υ 1 1 d <f <f d High-spin

\\ \\

111 111 1 d U

\ Low-spin

ill ill 111

Field Stabilization Energies, or LFSE's. The variation in LFSE across the transition-metal series is shown graphically in Fig. 8-6. It is no accident, of course, that theplots intercept the abscissa for d°, d5 and J10 ions, for that is how the LFSE isdefined. Ions with all other d configurations are more stable than the d°, d5 or d]0ions, at least so far as this one aspect is concerned. For the high-spin cases, we notea characteristic 'double-hump' trace and note that we expect particular stabilityconferred upon d3 and d8 octahedral ions. For the low-spin series, we observe aparticularly stable arrangement for d6 ions. More will be said about these systemsin the next chapter.

LFSE's for tetrahedral species are computed in a similar manner. They arecompared with the results for octahedral systems in Fig. 8-7. No illustration ofLFSE's for low-spin tetrahedral ions is included here because, as noted in Chapter5, the much smaller values of Atet relative to Aoci ensures that pairing energies Palways outweigh the ligand-field terms in practice. The trends summarized in Figs. 8-6 and 8-7 arise inevitably from ligand-fieldtheory. The LFSE terms are additional to those arising from the repulsive effects ofthe d electrons discussed in Section 7.1. Both contributions arise simultaneously,their common origins lying in the unequal filling of the d-orbital subsets. To someextent, these effects interact in that the magnitudes of the LFSE terms dependdirectly upon the strengths of the ligand fields which themselves vary in the samequalitative manner as functions of the t2nge™ configurations. It is not correct, however,to argue that the variation of Dq with bond length is the only consequence of thesteric role of the d configuration. Both d electron density and bonding electrondensity are affected, as we have seen.152 8 Complex Stability and Energetics

8.4 Energy and Structural Consequences in Real Systems

We shall now look at a number of thermodynamic and structural variations in

transition-metal compounds which owe their origins, at least in part, to the propertiesof open d shells.

8.4.1 Hydration Energies of Transition Metal(n) Ions

In Fig. 8-8 are plotted hydration energies for the first row, transition-metal divalentions. Water is a fairly weak-field ligand and so all these [M(H2O)6J2+ species arehigh-spin. We observe a 'double-hump' variation in AHhyd but with respect to agently curving, upward sloping curve. The similarlity between the relevant plot inFig. 8-6 and the experimental data in Fig. 8-8 has long been claimed as a splendidvindication of ligand-field theory at large. And so it is - but the variations in theobserved data of Fig. 8-8 compound many trends simultaneously and it is instructiveto consider them all. The enthalpies of hydration plotted in this figure refer to the process in Eq. (8.8).

M2+(g) + aq -> [M(H20)6]2+(aq) (8.8)

On the left, the divalent metal ion is spherical with a J-electron configurationwhich is amply described as dn. On the right, the metal is engaged in sixoctahedrally disposed bonds and its J-electron configuration is best recorded ast2£e™. The electronic contributions to the hydration process refer, as usual, to theformation of the bonds and the attraction of electrons to the central metal, to the 8.4 Energy and Structural Consequences in Real Systems 153

0 1 2 3 4 5 6 7 8 9 1 0

dn

Figure 8-8. Hydration energies for divalent transition metal ions.

interaction of the d electrons with the bonds, and to the interaction of the d electronswith each other. We consider each of these in turn. (i) Once more, let us take the extreme view that bond formation in these Wer-nerian complexes involves overlap of appropriate ligand orbitals with the metal 4s(and 4p) but not with the 3d orbitals. We adopt this hueristic extreme only toemphasize the largely separate roles of the s+p versus d orbital sets. The metal 4s(and 4p) overlap with the ligand will increase across the transition-metal periodowing to the decrease in covalent or ionic radius of the metal that accompanies theincreasing value of Zeff as discussed in Section 7.1. This effect accounts for the mainunderlying trend in Fig. 8-8. However, as we saw Section 7.1, the metal radii varyunevenly because of the steric effects of the t2^e™ configuration and so this effectalready contributes to the 'double-hump' form of the experimental plot in Fig. 8-8. (ii) Then we have the 'double-hump' contribution from the LFSE. Quantitatively,LFSE's are predicted as multiples of the ligand-field parameter Dq. However, Dqitself varies across the period since the ligand-field strength is affected by theproximity of the bond orbitals and the d orbitals. One obvious factor here is that Dqshould increase with decreasing bond length; another is that Dq should decrease asthe 3d orbitals contract with increasing Zeff. Experimental Dq values for divalentand trivalent first row transition-metal hexaaqua ions are plotted in Fig. 8-9. Thesedata do seem to reflect those predicted trends somewhat. We must always rememberthat Dq values are a complex function of many factors, however, so that the trendsin Fig. 8-9 cannot be explained as simply as suggested here. Nevertheless, thedependence of the LFSE's upon Dq, as in Figs. 8.5-8.7, is not affected by ourability to predict the Dq values themselves.154 8 Complex Stability and Energetics

(iii) We must look at the contribution to the hydration energies which arises fromthe interactions of the d electrons with each other. Behind this, in fact, is therecognition that our derivation of LFSE's took no account of such interelectronrepulsion and exchange, for we implictly began the discussion at the strong-fieldlimit. There are various ways of including the Coulomb terms and the followingshould be instructive. Within the hydration process in Eqn. (8.8), a spherical ion M2+ becomes a(hydrated) octahedral ion, [M(H2O)6J2+. Part of the Coulomb energy of the free ionconcerns repulsion and exchange terms within the dn< configuration. This is replacedby equivalent repulsion and exchange terms within the t2nge™ configuration. Let usestimate the trends in these quantities separately. From the outset, we recognize that Zeff increases as we traverse the transition-metal period, thereby contracting the orbitals and increasing both the Coulomb andexchange integrals. Let us separate this general trend from the discontinuous oneswhich arise from the uneven filling of the d or t2geg shells. Here, we focus upon thelatter. a) Repulsion terms in the free ions. Plot (a) in Fig. 8-10 indicates the variation ind-d repulsion energy as the d shell is progressively filled across the period. It hasbeen constructed in recognition of the fact that all d electrons repel all others; theeffect is especially large when two electrons are obliged to share the same orbital. b) Exchange terms in free ions. These are shown in plot (b) of Fig. 8-10. Here,the occurrence of each pair of parallel spin electrons within the same d shell givesrise to an attractive exchange energy. The number of parallel-spin electron pairs inthe dn configurations varies as O, O, 1, 3, 6, 10, 10, 11, 13, 16, 20 for η = O to 10respectively. c) Repulsion terms in the high-spin octahedral ions. These are estimated as aboveand shown as plot (c) in Fig. 8-10. At this level of approximation, they are the sameas in the free ion. The fact that the d orbitals are no longer equi-energetic is of noconsequence here. 8.4 Energy and Structural Consequences in Real Systems 155

Box 8-1 A reminder about Coulomb and exchange integrals

Coulomb and exchange integrals arise in connection with the two-electron Coulomb operator e2/ri2. Coulomb integrals take the form:

7(1,2) = J a*(l)b*(2)l — \ a(l)b(2)dT{dr2 (8.9)

Γ 12

cce(a*a\ >e(b* b)21

= J -- —-- (8.10)

where a and b are the spatial parts of different orbitals. By grouping together those parts of the integrand that refer to the same variables - electrons 1 and 2 - as in Eq. (8. 10), the integral 7(1,2) takes the form of a Coulombic repulsion between the charge clouds of orbitals a and b separated by distances r12. /(1,2) is called a Coulomb integral because of that. However, because of the indistinguishability of electrons, we must also include an integral of the form:

£(1,2) = fl*(l)fc*(2) — a(2}b(\}d^dr2 (8.11)

ffe(a*b\-e(b*a)2 = J - - —- - (8.12)

On grouping functions of the same variable again, as in Eq. (8.12), we find K(I, 2) to describe something like the repulsion between Overlap charge clouds'. This has no classical parallel and the integral is simply called the 'exchange' integral, for that is how it arose quantum mechanically. Both J and K integrals are intrinsically positive and both vary in such a way that the closer together the electrons 1 and 2, the larger the relevant integral. As we are considering here a and b to be different d orbitals, and as such belonging to the same atom, the magnitudes of the appropriate J and K integrals will be roughly comparable. Furthermore, the energies of the high-spin ground states in the species under present consideration will have the form E = C1/ - C1K, with c} and C2 positive. So, although J and K are intrinsically positive, while the Coulomb term gives rise to an increase in energy (repulsion), the exchange term incurs a decrease because of the minus sign (attraction). The idea that electrons can, in part, attract each other is, of course, a purely quantum phenomenon, arising ultimately from the indistinguishability of like particles.

d) Exchange terms in the octahedral ions. These are shown as plot (d) in Fig. 8-10. Now, because the t2g and eg orbital subsets are nondegenerate, the numbers ofequivalent, parallel-spin, electron pairs are much less, as shown in Table 8-3. Within the first-order estimations made here, it is apparent that no change in d-drepulsion energy accompanies the hydration process. Second-order adjustmentswould, of course, take account of the change in mean J-orbital radius on complexformation. Let us agree to stop at the simple level of correction here. Overall,therefore, the significant Coulombic change on hydration concerns the loss ofexchange stabilization.156 8 Complex Stability and Energetics

e) The differences in exchange terms. Plot (e) in Fig. 8-10 represents the changein exchange energy due to the break up of degenerate parallel spins; once more, wehave only made 'first-order' estimates. Here we see, yet again, a 'double-hump'contribution to the hydration energy. But we must be careful to read the plot theright way round. Plot (e) shows an increasing loss of stability following parallel-spin, electron pair disruption. So, the contribution to hydration energies will followthe trend shown in Fig. 8-11. This is the trend to be added to the Zeff, bond lengthand LFSE variations. We may only guess their relative contributions. Figure 8-12presents a possible scenario.

Table 8-3. The number of pairs of degenerate electrons with parallel spins in free-ion dn andoctahedral t2nge™ configurations.

The plots A-F in Fig. 8-12 are defined as follows. A is the variation due to thechanging Zeff, B is that due to bond weakening from to the repulsive effects of thet2nge™ configuration, C is the LFSE plot of Fig. 8-6, D is a 'correction' of this for thevariation in Dq shown in Fig. 8-11, and E is the exchange term of Fig. 8-10. Thecombined contributions from A, B, D and E are plotted as F. Overall, therefore, wereproduce the form of the experimental enthalpy curve in Fig. 8-8. While satisfying,even this is not the last word. This is because we have made our estimates of thevarious energy contributions in terms of the strong-field limit. The hexaquocomplexes lie towards the weak-field end of the appropriate correlation diagrams.The weak-field ground term for a d2 octahedral complex, for example, is 3T18(F). Aswe saw in Chapter 3, this term mixes with the excited 3Ti8(P) term. The 3Ti8(F)term correlates in the strong-field limit with the configuration t2j while the 3Ti8(P)state correlates with t^e\. In an intermediate field therefore, we can describe theground wavefunctions as combinations of t2g and t^e\ with mixing coefficients thatdepend upon the relative magnitudes of interelectron repulsion and the octahedralligand field. It is certainly possible to correct our view of hydration energies inorder to take all this into account: however, once again, we have no simple way ofgauging the relative importance of the contributions described in Fig. 8-12. So let 8.4 Energy and Structural Consequences in Real Systems 157

a,c

a =c e = d-b

Figure 8-10. A comparison of Coulombic and exchange contributions in the free-ion and high-spin octahedral complexes (see text).

contribution to ΔΗ hyd

Figure 8-11. Contributions to A// hyd due to disruption of numbers of degenerate, parallelelectrons.158 8 Complex Stability and Energetics

us be content* to recognize the many factors that are important while observing thatthe overall thermodynamic results of Fig. 8-8 are comprehensible, in principle, interms of these factors. Let us remember also that an explanation of the facts of Fig.8-8 is not simply forthcoming from the LSFE plot of Fig. 8-6 alone.

8.4.2 Lattice Energies of MCl2 Species

In Fig. 8-13 are plotted lattice energies for MCl2 species. The metal ions are high-spin and lie in octahedral sites in the lattice. The 'double-hump' form of the curveis obviously similar to that for the hydration energies we have just discussed. Thereasons for the observed trend in lattice energy are virtually identical to thosedescribed for hydration energies. In one system, a metal(n) ion is octahedrallycoordinated by six water molecules within a liquid medium; in the other, a metal(n)ion is octahedrally coordinated by six chlorine atoms within a solid lattice.

* Included in these would be the contributions that arise because of the different radial extentsof the d orbitals in the free ion and complex. The factors (a) and (c) in Fig. 8-10 will not cancelexactly but, because of their general form, this 'correction' is not expected to grossly modifythe qualitative form of plot F in Fig. 8-12 . 8.4 Energy and Structural Consequences in Real Systems 159

<D

.8

0 1 2 3 4 5 7 8 9 10 dn

Figure 8.13. Lattice energies for the formation of MCl2 compounds.

8.4.3 The Spinels

Spinels are compounds containing metals and group 16 elements of general formulaeAB 2 X 4 . The X2~ anions form an almost perfect cubic close packed array in which Xmay be O, S, Se or Te. By far the most numerous are the oxide spinels AB2O4.Spinel itself is MgAl2O4. The unit cell comprises 32 oxygen atoms and the formulaA8B16O32. The A metal ions are found in H, iv or vi formal oxidation states and theB metal ions then have πι, π or ι oxidation states respectively. Again, the mostcommon spinels are oxide species in which A are divalent and B are trivalentmetals. The cubic close packed anion lattice generates twice as many tetrahedralholes as octahedral ones. In so-called 'normal' spinels, the trivalent cations occupyhalf of the octahedral sites and the divalent cations one eighth of the tetrahedralsites: A1B^O4. In the 'inverse' spinel structure, one half of the trivalent cationsoccupy tetrahedral sites, while the other half and all the divalent cations occupyoctahedral sites: B^AB)O4. Typical species having the 'normal' structure are CdAl2O4, ZnAl2O4, Mn11Al11^O4and Zn11Fe11^O4. Examples having the 'inverse' arrangement are Ni11Fe11^O4 andFe11Fe11^O4 (Fe3O4 or magnetite). The question arises as to what factors determinethe choice of 'normal' or 'inverse' structure. The most immediately obvious ofthese are size and charge. Generally, size considerations would predict that thesmaller trivalent cations should occupy the smaller tetrahedral holes rather than thelarger octahedral ones. Charge considerations, on the other hand, predict that greaterlattice energies would result when the more highly charged cations are surroundedby the greater number of anions defining an octahedral hole. Straightaway, therefore,we see a conflict between these two factors and any real spinel structure involves abalance between the two. The charge factor appears to dominate in the 'normal'160 8 Complex Stability and Energetics

spinels. What has all this to do with the d shell and ligand-field theory? Well, theinversion of the structure in spinels like NiFeO4 is widely cited as a manifestationof the role of LFSE. The argument goes as follows. LFSE's for high-spin octahedral and tetrahedral species were plotted in Fig. 8-9.As a direct result of the smaller values of A tet relative to Aoct, the magnitude of thestabilization energies are significantly less for the tetrahedral than for the octahedralcomplexes. Other things being equal, we deduce that LFSE's favour octahedralover tetrahedral coordination, except in d°, d5 or J10 cases where both octahedraland tetrahedral LFSE's are zero. Now recall the balance of factors controlling 'nor-mal' and 'inverse' spinel structures above and let us supplement them with thenotion that the LFSE favours octahedral coordination for transition-metal ions otherthan those with J°, d5 or J10 configurations. We observe the 'normal' structure forZn11Fe11^O4 and Mn11Al11^O4, for example, and in these systems the transition metalsFe111 and Mn11 are d5 species. On the other hand, in Ni11Fe11^O4, while Fe111 is a d5ion, Ni n is J8. The LSFE factor therefore favours occupancy of the octahedral sitesby the nickel ions. NiFe2O4 is indeed observed to adopt the 'inverse' spinel structure,(Fe0^y(NiFe1 ^)0O4. A similar situation is observed in Fe11Fe11^O4 in which the d6iron(n) metal preferentially occupies octahedral sites. A more delicate balance occursin NiAl 2 O 4 in which a more nearly random site occupancy is observed:(AIo175Ni0125)XNi0-75Al1-75)0O4. The successful rationalization of these transition-metal 'inverse' spinel structuresin terms of the relative LFSE's of tetrahedral and octahedral sites is anotherattractive vindication of ligand-field theory as applied to structure andthermodynamic properties. Once again, however, we must be very careful not toextrapolate this success. Thus, we have a clear prediction that LSFE contributionsfavour tetrahedral over octahedral coordination, except for dn with η = O, 5 or 10.We do not expect to rationalize the relative paucity of tetrahedral nickel(n) speciesrelative to octahedral ones on this basis, however. Many factors contribute to this,the most obvious and important one being the greater stabilization engendered bythe formation of six bonds in octahedral species relative to only four bonds intetrahedral ones. Compared with that, the differences in LSFE's is small beer. 'Why',one asks, 'was our rationalization of spinel structures so successful when weneglected to include consideration of the bond count?' The answer is thatcancellations within the extended lattice of the spinels tend to diminish theimportance of this term. Recall that the unit cell in the spinels comprises A8B16O32. In the 'normal'structure, there are 16 B ions in octahedral sites and 8 A ions in tetrahedral ones.That corresponds to 96 octahedral B-O bonds and 32 tetrahedral A-O bonds or128 bonds in all. In the 'inverse' structure, we have 8 B ions in tetrahedral sites,8 B ions in octahedral ones, and 8 A ions in octahedral sites. This corresponds to 48octahedral B-O bonds, 32 tetrahedral B-O bonds and 48 octahedral A-O bondsor once again, 128 bonds in all. So the total number of M-O bonds, different typesto be sure, is the same in both 'normal' and 'inverse' spinel structures. We couldspend quite some time estimating the different bond energies of A-O and B-O or ofoctahedral versus tetrahedral, but that would undoubtedly involve a lot of guesswork.We can at least observe that the 'bond count' factor difference between the spinel 8.5 The Irving-Williams Series 161

structures is ameliorated compared with that involved with discrete octahedra andtetrahedra, and that this appears to bequeath to the LFSE a dominant role in thedetermination of spinel structures.

8.5 The Irving-Williams Series

In the late 1940's, Irving and Williams investigated the effect of varying the centralmetal ion on the stabilities of transition-metal complexes. Somewhat to theirsurprise, they found that a general pattern emerged, and that this pattern wasobserved both with a wide range of ligands and in the spectrum of biological activityassociated with transition-metal ions. They observed that for any given ligand, themagnitude of the stability constants varied along the series:

K(Mn) < AT(Fe) < K(Co) < K(Ni) < K(Cu) > K(Zn)

Some typical stability constant data are presented in Fig. 8-14, whilst Fig. 8-15shows some biological manifestations that illustrate the ubiquity of the effect. Thesequence is known as the Irving-Williams series. Can we rationalize these observations in terms of ligand-field or other effects?The data that we have presented in Fig. 8-14 refers to the log K1 values for eachligand with the high spin divalent metal ions. The sequence reflects a number ofsimple properties of the cations. Firstly, the trend closely parallels the 'ionic' radii

Figure 8-15. The biological activity of some transition-metal ions illustrating the Irving-Williams series.

of the metal ions ; as the metal ions decrease in radius, the metal-ligand interactionsincrease in magnitude and the stability of the complex increases. The downturn instabilities at the end of the series is associated with the increasing ionic radii. Thesecond correlation that we can investigate is with the LFSE associated with theelectronic configuration of the metal ion. These data are also presented in Fig. 8-16,and again there is some parallel with the stability constants; the larger the LFSE,the more stable the complex.

LFSE

IgKj for en

Fe Co Ni Cu Zn

Figure 8-16. Correlation of ionic radius and LFSE with log values for divalent transition-metal complexes of 1,2-diaminoethane. 8.5 The Irving-Williams Series 163

However, consideration in terms of the ionic radius or the LFSE shows that bothfactors predict that the maximum stabilities will be associated with nickel(n)complexes, as opposed to the observed maxima at copper(n). Can we give asatisfactory explanation for this? The data presented above involve KI values; and ifwe consider the case of 1,2-diaminoethane, these refer to the process in Eq. (8.13).

[M(H2O)6I2+ + en=^ [M(H2O)4(en)]2+ + 2H2O (8.13)

What happens if we look at the K2 or K3 values for didentate ligands? In general,

the K2 values show stability patterns which closely parallel those for KI. However,the K3 values are different. Figure 8-17 presents K3 data for transition-metalcomplexes of 1,10-phenanthroline and 1,2-diaminoethane (Eq. 8.14).

[M(H2O)6J2+ + 3L=^ [ML3J2+ + 6H2O (8.14)

The first feature that we note is the relative destabilization of the [CuL3J2+ complexescompared to the marked stabilization depicted in Fig. 8-16. The data for the 1,2-diaminoethane complexes now parallels the trends in ionicradius and LFSE rather closely, except for the iron case, to which we return shortly.What is happening? Copper(n) ions possess a d9 configuration, and you will recallthat we expect such a configuration to exhibit a Jahn-Teller distortion - the sixmetal - ligand bonds in 'octahedral' copper(n) complexes are not all of equalstrength. The typical pattern of Jahn-Teller distortions observed in copper(n)complexes involves the formation of four short and two long metal-ligand bonds.

Here is the true explanation for the position of copper(n) in the Irving-Williamsseries. When we consider the the replacement of water molecules by up to fourother stronger-field ligands, we expect the incoming ligands to form short (and thusstronger) copper-ligand bonds. The outcome is that the Jahn-Teller distortion resultsin shorter and stronger metal - ligand bonds than might be expected on the basis ofthe isotropic 'ionic radius' of copper(n). When we come to replace the remainingtwo water molecules, we form metal - ligand bonds which are considerably weakerthan expected. This is clearly seen when we consider the sequential data for theformation of copper(n) ammine complexes; the formation of pentammine andhexammine complexes is really very unfavourable. This is also reflected in oureveryday laboratory experience - the addition of excess concentrated ammoniasolution to copper(n) sulphate solutions results in the formation of the familiar deepblue solution containing the [Cu(NH3)4]2+ ion rather than [Cu(NH3)6]2+. This is furt-her emphasized when we compare the sequential log Kn values for copper(n) andnickel(n) ammonia complexes (Table 8-4). For the log K\, log K2, log K3 and log K4values we see the expected Irving-Williams pattern, with the copper(n) complexesbeing more stable than the nickel(II) complexes. However, when we come to thelog K5 and log K6 values, we see an inversion, with the nickel(II) complexes beingconsiderably more stable - indeed the value of log K6 for the copper(n) ammoniasystem cannot be measured in aqueous conditions.

Cu 4.2 3.5 2.9 2.1 -0.52

Ni 2.8 2.2 1.7 1.2 0.7 0.03

We further emphasize this destabilization of the fifth and sixth ligands binding tocopper(n) by considering the log KI, log K2 and log K3 values for 1,2-diaminoethanecomplexes (Fig. 8-18). Whereas the log K1 and log K2 data obey the Irving-Williamssequence, the log K3 parallel the trends in ionic radius and LFSE mentioned earlier.The data for tns(l,10-phenanthroline) complexes also illustrates the expected trend(Fig. 8-17). The anomalously high stability of the iron(n) complex can be readilyexplained when one considers that 1,10-phenanthroline is a very strong-field ligandand that the [Fe(phen)3]2+ cation is low-spin. The low-spin iron(n) centre is smallerthan the high-spin analogue (0.61 A as opposed to 0.78 A) and has a considerablygreater LFSE associated with it (24 Dq as opposed to 4 Dq). Our discussion of the Irving-Williams series illustrates, as ever, an importantgeneralization in transition-metal chemistry: in many cases there is no single, sim-ple principle which may be invoked to rationalize a given series of observations.Whilst LFSE effects are very important, they are but one of several factorscontrolling structure and thermodynamics. 8.5 The Irving-Williams Series 165

15 Ί

10-

ο-

-5 Mn Fe Co Ni Cu Zn

Figure 8-18. Stability constants for the formation of 1:1, 1:2 and 1:3 complexes with 1,2-diaminoethane.

Suggestions for further reading

Most of the standard texts mentioned in Chapter 1 have treatments of the materialdiscussed in this chapter. In many cases the approaches differ dramatically fromthat which we have adopted. Other texts which have relevant sections include:

In the preceding chapters we have developed a detailed understanding of the

behaviour of electrons in the d orbitals in transition-metal compounds. Can we nowuse this knowledge to rationalize some of the more familiar aspects of transition-metal chemistry? In this chapter we consider some of the chemical consequences ofthe J-electron configuration upon the chemistry of the transition metals. Some ofthe phenomena which we study are directly related to the number and arrangementof electrons in the d orbitals; others are indirectly related, being primarily dependentupon factors like the ionic radius. We address four main areas of interest to thecoordination chemist: coordination number and geometry, ligand choice, oxidationstate stability and rates of reactions.

9.2 Coordination Number and Geometry

It is reasonable to ask if it is possible to predict what the stoichiometry and geometry

of the product resulting from the interaction of a particular metal ion with aparticular ligand (or ligands) is likely to be. Can we make any progress towards thisgoal from our discussions in the earlier part of this book? As will becomeincreasingly clear, the answer is a mixed one: sometimes the interplay of d electronsin the valence shell is of prime and direct importance, sometimes of little importance,but more often it is relevant, yet only in an indirect way. The dominant features which control the stoichiometry of transition-metalcomplexes relate to the relative sizes of the metal ions and the ligands, rather thanthe niceties of electronic configuration. You will recall that the structures of simpleionic solids may be predicted with reasonable accuracy on the basis of radius-ratiorules in which the relative ionic sizes of the cations and anions in the latticedetermine the structure adopted. Similar effects are important in determiningcoordination numbers in transition-metal compounds. In short, it is possible to packmore small ligands than large ligands about a metal ion of a given size. This is most simply seen with monatomic ligands like the halides, which arereasonably approximated as spheres to which a meaningful radius may be assigned(F-, 1.19 A; Cl-, 1.67 A; Br, 1.82 A; I~, 2.06 A). We should preface this discussion

Transition Metal Chemistry. M. Gerloch, E. C. Constable

with the observation that, for intermediate oxidation states (+2 or +3) in donorsolvents, the vast majority of complexes are based upon a six-coordinate octahedralgeometry; aqueous solutions of divalent transition-metal salts all contain [M(H2O)6J2+ions. The octahedral geometry is favoured on electrostatic, ligand-field and packinggrounds. We choose manganese(n), a d5 ion which suffers no ligand-field imposedpreference for any particular geometry, to make the point. A considerable numberof complex manganese(n) fluorides, ranging from MnF2 itself through M[MnF3] toM2[MnF4], are known; all of these contain six-coordinate MnF6 units. In the relatedchloro complexes, the octahedral structure is also the most commonly encountered,although with some M cations, four-coordinate MnCl4 units are observed (rememberthat the radii of both the manganese and the other cation M are of importancewithin the crystal packing). In contrast, the bromo analogues exhibit a variety ofstructures in which MnBr6 and MnBr4 units are equally common. For example, bothK4[MnBr6] (containing a discrete [MnBr6]4" anion) and Cs2[MnBr4] (containing adiscrete [MnBr4]2' anion) are isolable compounds. Iodo complexes of manganese(n)are rather less common, but the majority appear to contain MnI4 tetrahedra. Similarpatterns exist with other d5 and J10 metal ions. For example, iron(n) forms the ion[FeF6]3" with fluoride, whilst both [FeCl6]3" and [FeCl4]' are formed with chloride,and only [FeBr4]" is known for bromide. Iron(m) iodo compounds are not commonlyfound since the iodide ion is a sufficiently strong reducing agent to usually reduceiron(in) to iron(n), with concomitant formation of iodine. A similar phenomenon isobserved in the reaction of iodide with copper(n) salts to give copper(i) iodide andiodine. With these d5 and J10 metal ions, the influence of the electronic configurationis only indirect, through the 'ionic radius' of the metal ion. One feature, exemplified above, is the tendency with 'borderline' ligands formetals to form stable four- and six-coordinate complexes. This is particularly markedwith J10 ions such as zinc(n), where the facile interconversion of four-, five- andsix-coordinate species is believed to play an important role in the biological functionof the metal in zinc metalloenzymes such as carbonic anhydrase. Further, it sohappens that with d1 ions such as cobalt(n) the ligand-field energies associated witha four-coordinate tetrahedral CoL4 complex and an octahedral CoL6 complex aresimilar. A common feature of cobalt(n) chemistry is the ready interconversion ofthese two coordination numbers. In both d1 and J10 ions, the absence of anyimportant ligand-field 'preference' for a particular coordination number or geometryis reflected in this facile interconversion between coordination types. A familiarexample of this is seen when an aqueous solution of cobalt(n) chloride isconcentrated. The (dilute) pink solution contains octahedral [Co(H2O)6J2+ ions. Uponconcentrating, the effective concentration of chloride increases and the solutionturns blue as the [CoCl4]2" ion is formed. The pink colour is recovered upon dilutionwith water. This colour change is the basis of a very simple 'invisible' ink, amessage written with dilute cobalt(n) chloride solution being invisible until thepaper is warmed. The replacement of zinc(n) by cobalt(n) in zinc metalloproteins isa trick commonly used by bioinorganic chemists. The ionic radii of the two metalsare somewhat similar, as is the tendency to undergo easy changes in coordinationnumber. The success of the strategy is seen in the observation that the cobalt(n)metalloproteins very frequently show activity similar to (occasionally greater than!) 9.2 Coordination Number and Geometry 169

the native zinc compounds. Why should this metal-ion exchange be useful? Zinc(n)is a d10 ion with no useful magnetic (diamagnetic) or spectroscopic (no 'd-d'transitions) properties, whereas cobalt(n) is a d1 ion with the associatedparamagnetism and 'd-d' spectra. Size effects are probably most readily illustrated with the highly structured ligandswhich characterize contemporary coordination chemistry. The concept of 'cone-angle' was originally developed by Tolman to explain some of the features ofphosphine coordination chemistry. The cone angle, 0, was, at its simplest, definedas the angle subtended at a nickel centre (defined by a Ni-P distance of 2.28 A)between vectors extending from the metal forming a tangent with the van der Waalsextremities of the substituents on the phosphine (Fig. 9-1).

Figure 9-1. Definition of the cone angle, 0, for trimethylphosphine.

Let us consider one specific example of how the cone-angle concept provides agood rationalization of the observed data. The reaction of nickel(n) bromide withPEtPh2 gives a complex [Ni(PEtPh2)2Br2]. This complex may be obtained as greenparamagnetic or red diamagnetic forms. The two forms may be interconverted and,in solution, dynamic equilibria are set up between the two. The red form has planargeometry and the green one, tetrahedral. There is an interplay of the ligand field ofthe relatively strong-field P2Br2 donor set, which favours the formation of the square-planar complex, with the steric repulsions between the relatively bulky phosphineand halide ligands, which favour the adoption of the tetrahedral geometry in which170 9 Chemical Consequences of the d-Electron Configuration

the various ligands are kept as far apart as possible. The cone angles, 0, of anumber of typical phosphines are listed in Table 9-1. If the PEtPh2 phosphine isreplaced by PEt3, with a smaller cone-angle, the percentage of the square-planarcomplex present at equilibrium increases, whereas replacement by PPh3 increasesthe percentage of tetrahedral form present at equilibrium. Similarly, the complexesprepared from nickel(n) chloride favour the square-planar forms, whereas thosefrom nickel(n) iodide favour the tetrahedral. In the solid state, the complexes withsmall cone-angle phosphines tend to be square-planar whereas those with the largercone-angle ligands (generally PAr3; Ar = aryl) tend to be tetrahedral.

Table 9-1. Cone angles for a series of phosphine ligands.

Phosphine θ

PMe3 118PEt3 132PEt2Ph 136PEtPh2 140PPh3 145P(C6Hn)3 170

Using extremely bulky ligands such as the bis(trimethylsilyl)amido anion

[(Me3Si)2N]-, it is sometimes possible to induce very low coordination numbers intransition-metal complexes. For example, a series of complexes like[Mi(Me3Si)2N)3] have been prepared. Other bulky ligands which have been usedinclude 2,6-di-tert-butylpyridine, tris(mesityl)phosphine and tris(2,6-di-tert-butylphenyl)phosphine. The use of such bulky ligands to stabilize low coordinationnumbers or to stabilize highly reactive centres (by 'shielding' them from reaction)is now well developed. In general, such effects are related purely to the steric bulk nof the ligands, and, once again, the only relationship to the d configuration arisesthrough the effective ionic radius of the metal center.* A further example of ligand control of the coordination number and geometry ofa complex is the use of relatively rigid polydentate ligands. For example, thephthalocyanato ligand 9.1 imposes a square-planar tetradentate N4 donor set onto ametal ion, and many metals form square-planar complexes with this type of ligand.Note, however, that some metal ions also coordinate another one or two axial ligandsto give square-based pyramidal or octahedral complexes. Similarly, the macrocyclic ligand 9.2 is expected to impose a planar pentagonalN5 donor set onto a metal ion. Although metal ions such as lithium form pentagonal

* A number of attempts have been made to rationalize the detailed preferences of particular dnconfigurations for certain geometries using molecular orbital and ligand-field based arguments.These arguments are beyond the scope of this book, and are not of general applicability to'normal' ligands and 'normal' oxidation state metal ions. 9.2 Coordination Number and Geometry 111

It is probably true to say that we are still unable to predict the number of a givenligand which will bind to a given metal ion. However, once we know the number ofligands that bind, we may use the Kepert approach (see Chapter 1) to accuratelypredict the spatial arrangement of these ligands. The one exception to both of thesepoints arises with metal ions which have a J8 configuration. We saw in Chapter 7 how the d% configuration can stabilize the square-planararrangement of four ligands about a metal center. This is the one real success thatwe bring to this discussion of coordination number and geometry. In the case offirst row transition-metal ions such as nickel(n), it is only very strong-field ligandswhich are capable of giving the necessary stabilization. We saw that weak-fieldligands such as chloride give tetrahedral anions like [NiCl4]2", whereas strong fieldspecies such as cyanide give square-planar ions like [Ni(CN)4]2". As we descend atriad, however, the ligand-field splittings increase, with the result that nearly allpalladium(n) and platinum(n) complexes are four-coordinate square-planar species.Again, note that this is not the case for nickel(n), where the majority of complexespossess six-coordinate octahedral geometries. A minor success is also seen in complexes of d9 and d4 ions, in which thedistorted octahedral geometries observed may be rationalized (and indeed predicted)in terms of the Jahn-Teller effect, and ultimately in terms of the steric activity ofthe open d shell. This is a common feature in copper(n) chemistry, and you will172 9 Chemical Consequences of the d-Electron Configuration

recall that it was a component in our explanation of the position of copper(n) in theIrving-Williams series. The final comment we shall make in this section concerns the formation ofcomplexes with the low coordination numbers two and three. We remarked abovethat the existence of such complexes can be favoured by the use of stericallydemanding ligands like the bis(trimethylsilyl)amido anion. Such ligands tend to beassociated with transition metals to the left of the series. A second group of metalions, but to the extreme right of the transition series, is also found to form a rangeof two and three coordinate complexes. This can be partially explained onelectrostatic grounds. As we place more and more electrons into the d manifold ofthe metal ion, the interelectronic repulsions between the metal ion and the ligandsincrease. With a full d shell, these repulsions are sufficient that a range of two andthree coordinate complexes like [Ag(CN)2]" and [CuCl3]2" are found with lowoxidation state J10 ions. This observation has been further rationalized in terms ofthe relative energies of the ns, np and (n-l)d orbitals. However, it should be notedthat these low-coordinate complexes are only associated with the lower oxidationstates. Zinc(n) exhibits the usual range of four- and six-coordinate complexes.

9.2.1 Coordination Numbers in Low Oxidation State Complexes

We have seen that complexes in low formal oxidation states (+1, zero or negative)can be stabilized by the use of strongly ^-acceptor ligands like carbon monoxide oralkenes. The stabilization is associated with an increase in the ligand-field splittingresulting from the overlap of the t2g set of d orbitals with the π levels of theligands. We also noted that these low oxidation state compounds with π-acceptorligands are more covalent, and that the large A values resulted in a markedstabilization of compounds in which only the nine lowest-lying molecular orbitalsare occupied - the so-called eighteen electron rule. Where the eighteen electronrule is obeyed, we can predict the number of ligands within a particular lowoxidation state compound. In many respects, the successes of this model are remarkable. Iron(O) possesses atotal of eight electrons in its valence shell. To satisfy the eighteen-electron rule,five two-electron donors are needed, and compounds such as [Fe(CO)5] are formed.These molecules also obey simple VSEPR precepts, and [Fe(CO)5] adopts a trigo-nal bipyramidal geometry. Conversely, the use of two five-electron donor ligandssuch as the strong ^-acceptor cyclopentadienyl, Cp, gives the well-known compoundferrocene (9.3).

9.3 9.3 Ligand Types - The Concept of Hard and Soft 173

In a similar vein, we observe nickel(O), possessing ten electrons in its valence shell,to require four carbonyl ligands to satisfy the eighteen electron rule and form[Ni(CO)4], whilst chromium(O), with six electrons in its valence shell forms[Cr(CO)6]. These latter compounds are tetrahedral and octahedral respectively. What happens if the metal center possesses an odd number of electrons in thevalence shell? Manganese(O) has seven electrons in its valence shell, and requires atotal of 11 electrons to satisfy the eighteen electron rule. We can 'cheat' by usingthe five electron donor, Cp, and three carbonyl ligands, as in the compound[(Cp)Mn(CO)3] (9.4). What happens if we only have carbonyl ligands? We couldform either [Mn(CO)5] (17 electrons) or [Mn(CO)6] (19 electrons), although weknow that the latter species is particularly unfavourable. In fact, two things canhappen. The 19 electron species [Mn(CO)6] 'wants' 18 electrons in the valenceshell, and can achieve this by the loss of an electron to form the cation [Mn(CO)6J+.The alternative is for the Mn(CO)5 units to dimerize, with the formation of a two-center, two-electron Mn-Mn bond in [Mn2(CO)10] (9.5) and so achieving an 18electron configuration about each metal center.

8 ,,CO 8I ^CO OC Mn Mn—CO C(T CO δ 9.4 9.5

When using the eighteen electron rule, we need to remember that square-planarcomplexes of J8 centers are associated with a 16 electron configuration in thevalence shell. If each ligand in a square-planar complex of a J8 metal ion is a two-electron donor, the 16 electron configuration is a natural consequence. Theinterconversion of 16-electron and 18-electron complexes is the basis for the modeof action of many organometallic catalysts. One of the key steps is the reaction of a16 electron complex (which is coordinatively unsaturated) with a two electron donorsubstrate to give an 18-electron complex.

9.3 Ligand Types - The Concept of Hard and Soft

One of the features of coordination chemistry which we try to explain is the

'preference' of certain metal ions for certain ligand types. In this section, we brieflydiscuss the models which have been developed to rationalize observed patterns ofligand recognition. We preface our discussion by noting, however, that the models174 9 Chemical Consequences of the d-Electron Configuration

Box 9-1 The common ligands in low oxidation state (organometallic) chemistry and the number of electrons which they donate to the metal are indicated below. Note that the electron counting scheme that we use treats all groups as neutral. In other words, we start from neutral metal centers and treat formally anionic organic fragments as radicals. Further explanation of this point is to be found in some of the suggestions for further reading.

Ligand Number of electrons

alkyl 1 aryl 1 hydride 1 carbonyl 2 alkene 2 phosphines 2

allyl 3 (or more rarely, 1)

diene

cyclopentadienyl 5 (or more rarely, 3 or 1)

arene ArH 6

The reader is left to determine the valence shell electron count for each of the following molecules or ions: [MeMn(CO)5], [Co2(CO)8], [PhCr(CO)5]-, [(C6H6)Mo(CO)3], [(C6H6)JVIo], [Cr2(CO)10]2-, [ReH9]2-, [(H2CCHCH2)Mn(CO)5], [(H2CCHCH2)Mn(CO)4], [(C4H6)Fe(CO)3], [Ti(Cp)4], [Ni(Cp)2] and [Ni(H2CCHCH2)2]

have very little to do with the J-electron configuration of the transition-metal ion,and more to do with the ionic size and the charge on the ion. We begin by considering the stability constants for the formation of halidecomplexes with zinc(n) and mercury(n) (Table 9-2) Notice that the stability of the zinc complexes decrease as F > Cl > Br > I, atrend that is exactly reversed for mercury. Some metals - Class (a) - form complexes

Fe(IIi) 6.0 1.4 0.5

Pb(Ii) 0.3 1.0 1.2 1.3

Mn+ + X-

Ligands which form stronger complexes with Class (a) metals are described as hard and those which form stronger complexes with Class (b) metals are called soft. Hard metals form more stable complexes with hard ligands and soft metalsform more stable complexes with soft ligands. A listing of hard and soft metals and ligands is presented in Table 9-4.

Fe(ii), Co(ii), Ni(ii), CU(H), Rh(m), Ir(m), RU(III), Os(n)

Hard metal ions are either highly charged and/or relatively small with a highcharge to radius ratio. This results in the valence shell electrons being stronglybound to the metal and less available for entering into covalent bonding with aligand. Consequently, interaction with ligand donor atoms having highelectronegativities is favoured. Hard -hard interactions are more electrostatic. Incontrast, soft metal centers have low charge to radius ratios and interact with lesselectronegative donor atoms. Soft -soft interactions are frequently more covalent incharacter. Of course, many of the ions of interest to a transition-metal chemist are'intermediate' in character - and might do anything!176 9 Chemical Consequences of the d-Electron Configuration

9.4 The Stabilization of Oxidation States, and

Reduction Potentials

9.4.1 Reduction Potentials and Thermodynamics

In the introductory chapter we stated that the formation of chemical compounds

with the metal ion in a variety of formal oxidation states is a characteristic oftransition metals. We also saw in Chapter 8 how we may quantify the thermodynamicstability of a coordination compound in terms of the stability constant K. It isconvenient to be able to assess the relative ease by which a metal is transformedfrom one oxidation state to another, and you will recall that the standard electrodepotential, E®, is a convenient measure of this. Remember that the standard freeenergy change for a reaction, AG0, is related both to the equilibrium constant (Eq.9.1) " (9.1)and to the standard electrode potential (Eq. 9.2) AG0 = -zFE® (9.2)where ζ is the number of electrons involved in the redox process. From Eqs. (9.1)and (9.2), we obtain the relationship between the standard electrode potential andthe stability constant for a redox process as shown in Eq. (9.3). E® = (RT/zF)lnK (9.3) We can thus use E® values to gauge the effects that various ligands have uponthe stability of one given oxidation state with respect to any other.

9.4.2 Intermediate Oxidation States

Negatively charged ligands are expected to stabilize higher oxidation states, and wewill probe such effects shortly. Meanwhile, we may eliminate the effects of chargedligands (but not, of course, of different dipoles within a ligand) by comparingcomplexes with neutral ligands. Consider the cobalt complexes with six water (Eq.9.4) and six ammonia ligands (Eq. 9.5). [Co(H2O)6P+ + e- -> [Co(H2O)6J2+ E® = +1.84V (9.4) 3+ 2+ [Co(NH3)6] + e- -» [Co(NH3)6] E® = +0.1OV (9.5) The reduction potentials indicate* that the cobalt(m) aqua complex is unstablewith respect to the cobalt(n) state, whereas the cobalt(m) ammine complex is

*Remember that the relevant potentials to consider are there for the oxidation and reduction ofwater. 9.4 The Stabilization of Oxidation States and Reduction Potentials 111

stabilized. In aqueous solution, we need to consider the various redox processes by

which water itself maybe oxidized or reduced-for example, [Co(H 2 O) 6 J 3+ WiIl oxi-dize water generating dioxygen and cobalt(n). Note that the large positive E^for thereaction with water ligands indicates that [Co(H2O)6]3+ is not likely to be an isol-able species in water and, in practice, such salts may only be obtained with thegreatest difficulty. Ammonia is a stronger-field ligand than water (/values of 1.25and 1.0 respectively). Cobalt(n) is a d1 ion whereas cobalt(m) has a d6 configuration.The somewhat stronger ligand-field of the six ammonia ligands is enough to stabilize 6 3+the low spin d configuration in the [Co(NH3)6] ion with its associated large LFSE.It is interesting to note that very few high-spin cobalt(m) complexes are known, andthose that are possess negatively charged ligands. It is evident that in order to sta-bilize cobalt(m), it is necessary to have ligands which produce a sufficiently strongfield to overcome the pairing energies associated with the formation of a low-spinconfiguration. We may usefully imagine the oxidation process to occur in two stages.Firstly, the rearrangement of the high-spin d1 cobalt(n) ion to a low-spin t^e\configuration and, secondly, the removal of an electron from the eg orbital of thecobalt(n) ion. The larger the ligand-field splitting, the greater the stabilization ofthe low-spin d6 cobalt(m) complex. It is this ligand-field stabilization of the d6 ionwhich compensates for the unfavourable electron repulsion associated with the lowspin configuration. The subtlety of the effects involved in determining the observedE& values is illustrated further by the cobalt 2,2'-bipyridine complexes (Eq. 9.6).

[Co(bpy)3]3+ + e--> [Co(bpy)3]2+ ΕΘ= +0.31V (9.6)

Here we focus upon two competing effects. The 2,2'-bipyridine is a strong-field

ligand (f = 1.33) which will give large ligand-field splittings for both the cobalt(n)and cobalt(m) complexes. As we saw for the ammonia complexes, the splitting issufficiently large to stabilize a low-spin cobalt(m) state. If this were the onlyimportant contribution, we would expect 2,2'-bipyridine to stabilize the cobalt(m)state more than ammonia does (since it is a stronger field ligand than ammonia).However, 2,2'-bipyridine is a π-acceptor ligand (indeed, this is the reason for itsposition in the spectrochemical series). The large splittings in the 2,2'-bipyridinecomplexes arise from the interaction of filled t2g orbitals on the metal with the π*orbitals of the ligand and the resultant lowering of the energy of the t2g set as shownin Chapter 6. The electron rich d1 cobalt(n) ion is a better ττ-donor than the higheroxidation state d6 cobalt(n) ion. Thus, the resultant lowering of the t2g orbitals willbe more effective with the cobalt(n) than the cobalt(m) ion. The balance is such thatthe stabilization of the cobalt(m) state is less with 2,2'-bipyridine than with theweaker-field NH3 ligand! The paradox is that we would normally expect largerligand-field effects to be associated with the higher oxidation state. We investigatethis in a little more detail by studying some iron(ii)/(m) complexes in which the d6configuration is associated with the lower oxidation state.

[Fe(H2O)6I3+ + e- -> [Fe(H2O)6I2+ EQ = +0.77V (9.7)

[Fe(bpy)3]3+ + e~ -» [Fe(bpy)3]2+ EQ = +0.97V (9.8)

178 9 Chemical Consequences of the d-Electron Configuration

Consider, then, the iron complexes with water (Eq. 9.7) and 2,2'-bipyridine ligands(Eq. 9.8). The larger E® with the 2,2'-bipyridine ligand indicates a greater stabili-zation of the iron(II) complex with the stronger-field ^-acceptor ligand. This is forexactly the same reasons we discussed when comparing the cobalt complexes. Inthis case, the lower oxidation state is stabilized by the 2,2'-bipyridine ligand. Thisis because of the favourable ligand-field terms associated with the d6 configurationwith the strong field ligand. In the iron complexes, this stabilizes the iron(n) statewhereas in the cobalt complexes, it is the cobalt(m) state which benefits and in fact, 3+the blue [Fe(bpy)3] ion is not particularly stable in water. Even this is not thewhole story for we are not really comparing like with like - the 2,2'-bipyridinecomplexes of iron are low-spin in both the iron(n) and iron(in) states whereas theaqua complexes are high spin. So we see the origin of an additional stabilization ofthe d6 [Fe(bpy)3]2+ complex ion. As expected, the introduction of negatively charged ligands results in thestabilization of the higher oxidation states. This is seen most simply in thecomparison of aqua and oxalato complexes of cobalt (Eqs. 9.9 and 9.10). Oxalate iscomparable in ligand-field strength to water (f = 0.99) but the negatively chargedligands stabilize the higher oxidation state. All of our remarks regarding the changefrom the high-spin cobalt(n) to the low-spin cobalt(m) ion pertain here.

[Co(H2O)6I3+ + e~ -> [Co(H2O)6J2+ EQ = +1.84V (9.9)

A similar stabilization of the cobalt(m) state is observed if we use a chelating

ligand such as edta4~ (Eq. 9.11), which completely 4 wraps-up' the metal center.Here is a most important point. Why do negatively charged ligands stabilize higheroxidation states? It has more to do with the entropy term associated with thesolvation of the more highly charged ions than the enthalpy term reflecting anydifferences in M-L bond strengths. What happens if we 'boost' the effect of negatively charged ligands by choosingone which is also a strong-field π-acceptor? A good example is provided by thecyano complexes (Eq. 9.12). Note that in this case, the high-spin square-basedpyramidal cobalt(n) ion is only coordinated to five cyanide ligands.

[Co(CN)6]3- + e--> [Co(CN)5]3- + CN~ E& = -0.83V (9.12)

This may be rationalized in terms of two factors. Firstly, the tendency to build up alarge charge density on the cobalt(n) center would be great (but remember theelectroneutrality principle) and, secondly, the ligands are labilized by the presenceof electron density in the eg orbitals of the high-spin cobalt(n) ion. The loss of onenegatively charged ligand is not possible in the chelated oxalato or edta complexes(Eqs. 9.10 and 9.11). The massive stabilization of the cobalt(m) state is partiallydue to the negatively charged ligands and partly due to the ligand-field stabilizationof the low-spin d6 ion. 9.4 The Stabilization of Oxidation States and Reduction Potentials 179

Similar effects are observed in the iron complexes of Eqs. (9.13) and (9.14). Thecharge on the negatively charged ligands dominates the redox potential, and weobserve stabilization of the iron(ni) state. The complexes are high-spin in both the 6oxidation states. The importance of the low-spin d configuration (as in our 4discussion of the cobalt complexes) is seen with the complex ions [Fe(CN)6] ' and 3[Fe(CN)6] - (Eq. 9.15), both of which are low-spin. 3 4 & [Fe(Ox)3] - + e- -> [Fe(ox)3] - E = +0.02V (9.13)

[Fe(edta)]- + e~ -» [Fe(edta)]2- ΕΘ = -0.12V (9.14)

[Fe(CN)6]3- + e- -» [Fe(CN)6]4- E® = +0.36V (9.15)

The now-familiar balance of effects operates. The anionic ligands favour thehigher oxidation state (again, associated with a solvation effect), the ligand-fieldstabilization of the low-spin d6 iron(n) center is considerably greater than that ofthe low-spin d5 iron(m) center, and back-donation is expected to be greater in theiron(n) complex. The latter is indeed the case for the Fe-C distances in the iron(n)compound are slightly shorter than those in the iron(m). The overall balance is astabilizing of the iron(m) state with respect to complexes with aqua ligands, incontrast to those with the neutral strong-field ligand bpy.

9.4.3 The Electroneutrality Principle - A Reprise

In Chapter 1 we introduced the electroneutrality principle. We now consider some

of its implications. You will recall that we described the [Fe(H2O)6J3+ ion as 50%covalent (or ionic). A similar description of 50% covalency may be applied to thecobalt(III) complex ions [Co(H2O)6J3+ and [Co(NH3)6]3+. The higher the oxidationstate, the greater the covalency necessary in the bond to fulfill the requirements ofthe electroneutrality principle. The electronegativities of cobalt(m), N and O are2.0, 3.0 and 3.4 respectively. The smaller difference in electronegativities betweencobalt(m) and nitrogen than between cobalt(m) and oxygen means that the Co-Nbond will be more covalent than the Co-O bond. In accord with the requirementsof the electroneutrality principle, [Co(NH3)6]3+ will be more favoured than[Co(H2O)6J3+. This is one of the observations that we discussed in the previoussection. Note that we have come to the same conclusion without invoking anyknowledge of the number or arrangement of the d electrons. Let us extend our discussion of the [Fe(H2O)6J3+ cation a little further. Theelectronegativities of oxygen and hydrogen are 3.4 and 2.2 respectively, and theO-H bond should thus be polarized in the sense H^-Οδ". The electroneutralityprinciple applied to the Fe-O interaction resulted in our placing half-positivecharges upon each of the oxygen atoms. If we now consider the H-O interactions,we may reallocate charges in accord with the electronegativities and theelectroneutrality principle such that the positive charges reside on the hydrogens.Charge neutrality of the oxygen would then establish a final charge distribution180 9 Chemical Consequences of the d-Electron Configuration

which places a quarter positive charge on each hydrogen atom, and each oxygenatom and the iron center are neutral. The overall charge on the complex ion (12 χ1/4 = 3) Is thus, of course, unchanged. 2+ Consider the closely related ion [Fe(H2O)6J . The only difference is in the for-mal oxidation state of the metal ion. If an ionic model is assumed (9.6), the chargeon the metal center is +2. A purely covalent model results in the placing of a formalquadruple negative charge upon the iron center (9.7). To satisfy the electroneutralityprinciple, and establish a near-zero charge on the metal, each oxygen atom is

OH9 OH2 OH7 OH2 H2O,. 4-

OH9 OH2

OH9 + OH2

9.6 9.7

required to donate 1/3 of an electron (9.8). Pauling describes this situation as being33% covalent (or 66% ionic). We could envisage further distribution of the electroniccharge such that both the iron and the oxygen atoms are neutral, so giving a 1/6positive charge associated with each hydrogen atom. We shall return to thisobservation shortly. Remember, meanwhile, that there is a smaller positive chargeassigned to the hydrogen atoms in [Fe(H2O)6J2+ than in [Fe(H2O)6J3+.

1/3+ OH2 1/3+

,-OH9

+ OH2

9.8

Now consider an iron(m) complex with six negatively charged ligands. The purelycovalent representation places a -3 charge on the metal center and leaves eachligand neutral. The electronegativity principle makes the iron neutral and places ahalf-negative charge upon each ligand. Ligand donor atoms are invariably moreelectronegative than metal centers, and so this distribution of charges is favoured.Compare this situation with that in the [Fe(H2O)6J3+ ion in which we ended upplacing positive charge upon the ligands. This phenomenon, which ultimately restsupon the typical relative electronegativities of the metal and ligands, provides asecond main cause of the stabilization of higher oxidation states by negativelycharged ligands. 9.4 The Stabilization of Oxidation States and Reduction Potentials 181

9.4.4 Protic Equilibria Involving Coordinated Ligands

We have just noted that the electroneutrality principle suggests that, in cationicaqua complexes, the hydrogen atoms of the water ligands acquire positive charge.Furthermore, we observed that the higher the oxidation state, the greater the positi-ve charge and the greater the polarization of the O -H bonds. In other words, thehigher the oxidation state, the more acidic the water ligands become and equilibriaof the type shown in Eq. (9.16) become accessible.

[L5M(OH2)]"^ [L5M(OH)]^+ + H+ (9.16)

water molecules are relatively acidic (Table 9-5). Water coordinated to an iron(m)center is a stronger acid than acetic acid!

Table 9-5. pKa values for coordinated water molecules.

H2O 15.6[Al(H2O)6I3+ 5.0[Fe(H2O)6J3+ 2.0 2+[Zn(H2O)6I 9.5

If we combine this observation with the previous discussion regarding the use ofnegatively charged ligands to stabilize higher oxidation states, we have a self-regulating way in which aqua ions may 'adjust' their coordination environment asthe oxidation state of the central metal ion changes. The higher the oxidation stateof the metal ion, the greater the polarization of the water molecule and the moreacidic it becomes; the more acidic the water, the greater the tendency to formhydroxide (or even oxide) ligands which then stabilize the high oxidation state ofthe metal ion. Consider some vanadium ions in aqueous solution. Pale violet solutions of 2+vanadium(n) salts contain the [V(H2O)6J ion. The vanadium(n) center is onlyweakly polarizing, and the hexaaqua ion is the dominant solution species. Aqueousvanadium(n) solutions are observed to be unstable with respect to reduction ofwater by the metal center. In contrast, vanadium(in) is more highly polarizing andan equilibrium between the hexaaqua and pentaaquahydroxy ion is set up. The pKaof 2.9 means that the [V(OH2)6]3+ ion (Eq. 9.17) only exists in strongly acidicsolution or in stabilizing crystal lattices. 3+ 2+ + [(H2O)5V(OH2)J ^ [(H2O)5Y(OH)J +H ρΚΆ =2.9 (9.17)

Vanadium(iv) is even more strongly polarizing. The first deprotonation process is

not observable in aqueous solution. The pentaaquahydroxy ion may be present in182 9 Chemical Consequences of the d-Electron Configuration

very strongly acidic solutions, but the dominant solution species is one which isderived from a second deprotonation of the pentaaquahydroxy complex. This couldgive rise to a bis(hydroxy)tetraaqua ion or an oxopentaaqua species. We will seelater that the multiply bonding π-donor oxo group stabilizes higher oxidation stateions, and it is, in fact, this latter species which is present (Eq. 9.18).

The blue [(H2O)5V( = O)]2+ ion is the vanadyl ion which is usually depicted asVO2+. Actually, the vanadium center is still sufficiently polarizing that a thirddeprotonation equilibrium is established in aqueous media to generate the ion[(H2O)4V(=O)(OH)]+, which contains water, hydroxy and oxo ligands (Eq. 9.19).

[(H2O)4V(^ O)(H2O)I2+^ [(H20)4V( = O)(OH)]+ + H+ ΚΛ 6.0

Ρ (9.19)

A similar situation pertains for iron salts in aqueous solution. Solutions of iron(n)salts contain the very pale green cation [Fe(H2O)6I2+, although these solutions oftenappear with various darker shades as a result of aerial oxidation. In the solid state,the alum KFe(SO4)2-12H2O is a very pale violet colour, and contains the [Fe(H2O)6J3+ion. Solutions of this compound or other iron(m) salts are usually varying shades ofyellow, although very pale coloured solutions may be obtained in acidic conditions.The yellow coloration is due to the deprotonated species which exhibit a ligand -metal-charge transfer transition in the ultraviolet region which tails into the visible.Both mono- (Eq. 9.20) and bis-deprotonated (Eq. 9.21) complexes are present inaqueous solution. Note the difference between the iron(in) and the higher oxidationstate vanadium(iv) complexes. In the latter case, an oxo ligand was generated afterthe second deprotonation to stabilize the high oxidation state metal centre, whereaswith the lower oxidation state iron(m) centre, a bis(hydroxy) complex is formed(Eq. 9.21).

[(H2O)5Fe(OH2)]3^ [(H2O)5Fe(OH)J2+ + H+ ρΚΛ = 2.0 (9.20)

[(H2O)4Fe(OH2)(OH)J2+^ [(H2O)4Fe(OH)2J+ + H+ pKa = 3.3 (9.21)

This leads us to a second aspect of the formation of hydroxy ligands in higher

oxidation state complexes. The deprotonation of a coordinated water ligand to ahydroxy ligand is frequently associated with the formation of polynuclear complexesin which the hydroxy ligands are associated with two metal ions which they bridge,as opposed to a single metal center. This is a process which is known as olation andwas first described by Werner in his pioneering studies of kinetically inert cobalt(m)complexes. The vanadium(m) ion [V(H2O)6J3+ (9.9) exhibits this behaviour, with a log K of 4associated with the formation of the hydroxy-bridged dinuclear complex 9.10. Thisis a general phenomenon. For example, chromium(m) and iron(m) form strictly 9.4 The Stabilization of Oxidation States and Reduction Potentials 183

— — 3+ - H2 ? OH2 -, 4+ ()H 2 H2C~\ TT I / ^OH2 P—-V OH2

Tx OTT UH2 ~ T ΗT M \ /" + 2H 2 O s/ υ / 2 "H OH2 2 OH2 OH2

9.9 9

analogous olated dinuclear complexes. In the case of the iron(m) species, the processcan proceed further to generate bridging oxy ligands (Eqs. 9.22-9.24).

[(H2O)4Fe(OH)2Fe(H2O)4I4+^ [(H2O)4Fe(O)2Fe(H2O)4I2+ + 2H+ (9.24)

This is only the beginning of a process which ultimately results in the formationof solid state hydroxides or oxides. Actually, the solution species present in neutralor alkaline solutions of transition-metal ions are relatively poorly characterized.The formation of numerous hydroxy- and oxy-bridged polynuclear species makestheir investigation very difficult. However, it is clear that there is a near-continuoustransition from mononuclear solution species, through polynuclear solution speciesto colloidal and solid state materials. By the way, the first example of a 'purely'inorganic compound to exhibit chirality was the olated species 9.11.

6+ NH3

ι/v H1N- -Co OH H ,Co NH3

HO- .Co-O HCT OH H3N—-Co NH3

N H3 NH3

9.11184 9 Chemical Consequences of the d-Electron Configuration

9.4.5 The Stabilization of High Oxidation States

There is an interesting paradox in transition-metal chemistry which we have

mentioned earlier - namely, that low and high oxidation state complexes both tendtowards a covalency in the metal - ligand bonding. Low oxidation state complexesare stabilized by ^-acceptor ligands which remove electron density from the electronrich metal center. High oxidation state complexes are stabilized by /r-donor ligandswhich donate additional electron density towards the electron deficient metal centre. The stabilization of high oxidation state compounds might seem particularlyparadoxical. The highest oxidation states are usually only stabilized by ligands suchas fluoride and oxide. Complex species such as [MnO3F] and [Mn2O7] represent thestabilization of manganese(vn), [CrO4]2~, [Cr2O7]2" and [CrO3Cl]" of chromium(vi),whilst iron(vi) is observed in Na2[FeO4]. The oxo ligands form formal double bondswith the metal, and the short M-O distances that result allow efficient transfer ofcharge to the electron deficient metal centre. In the case of fluoride, the short M-Fdistances (as fluoride is a relatively small ligand) allow efficient overlap betweenthe filled 2p orbitals of the fluorine and the empty orbitals of the metal. And yet afurther paradox: the ligands which stabilize the highest oxidation states are thosewith the most electronegative donor atoms! However, despite the fluorine beingelectronegative, it is acting as a ;r-donor to the metal!

9.4.6 The d Orbitals, Covalent Character and Variable Oxidation States

- A Summary

In Chapter 6, we introduced the idea of the variable role of the d orbitals in

transition-metal complexes as a function of changing oxidation state. At that point,we focused upon the difference between low oxidation states and 'higher' ones(meaning those typical of Werner-type compounds; say, +2 or +3). In this chapter,we have concentrated rather more on high oxidation states and have noted the'paradoxical' variations in the covalent character of the M-L bonds with varyingoxidation state. We now draw these various themes together and provide an overviewof changing bonding character throughout the J-block chemistry. In Fig. 9-2, we offer a schematic summary of the determinants of covalentcharacter in transition-metal bonding. In intermediate and high oxidation states, the (3)d orbitals are increasingly 'inner'with respect to the (4)s and (4)/? orbitals. As discussed repeatedly throughout thisbook, these d orbitals may be considered as largely uninvolved in direct overlapwith the ligand orbitals: they are essentially excluded from the metal's valenceshell. In changing from, say, the +2 oxidation state to the +7 state, the polarizingpower, or hardness, of the notional Mn+ ion increases dramatically and non-linearly(curve 1 in Fig. 9-2). Higher oxidation states will be accessible only with hardligands (soft anions would reduce the metal in these higher oxidation states). Thoughretained tightly by the ligands, their electron density is drawn towards the morehighly charged metal and so the covalent character of the M-L bonds increasessteadily with increasing n. The same conclusion follows with the recognition that 9.4 The Stabilization of Oxidation States and Reduction Potentials 185

. Polarizing power and

electronegativity of Mn

ailability of d orbitals 'erlap

Resultant covalent character

+7 +2

Oxidation state

Figure 9-2. The variation of covalent character with oxidation state.

the electronegativity of a progressively more positive metal ion rapidly approaches,

or even surpasses, that of even an atom like fluorine. So the (perhaps) 50% covalentcharacter of the M-L bonds in typical metal(n) complexes increases with increasingoxidation state. On the other hand, a decrease in oxidation state from metal(n) quite rapidlyreleases the (3)d orbitals into the valence shell. The d orbitals are full, or frequentlyso, so that M-L bonding electron density now derives from both metal and ligand(this is in contrast to the dative covalency of higher oxidation state complexes inwhich the electron density emanates from the ligands alone). We thus observe anincrease in covalent character on decreasing the oxidation state from metal(n), butthis time originating from the rapidly increasing participation of the (3)d orbitals inthe valence shell (and the emerging dominance of the 18-electron rule). Towardsthis extreme, we see the growing importance of soft metal/soft ligand interactions,mediated by the synergic 'back-bonding', first mooted by Chatt, Dewar andDuncanson. Throughout the scheme summarized above, we are to understand that multi-ple bonding involves (3)άπ- Ln overlap in the low oxidation state complexes, but(4^pn-Ln overlap in the high oxidation state complexes. These latter are generallycharacterized by substantially shorter bonds, thus facilitating ρπ-1^π overlap relativeto a presumably small such contribution in less tightly bound Werner-type systems.In any case, the degree of (4)pn-Ln overlap need not be great since all that isrequired to satisfy the electroneutrality principle is a sufficient drift of electrondensity towards the metal. This could be dominated by cr-bonding contributionsalthough we need not guess the relative proportions of σ and TT, however, to makethe main point.186 9 Chemical Consequences of the d-Electron Configuration

9.5 Consequences of the rf-Electron Configuration

upon Reaction Rates

Thus far, we have focused upon the thermodynamic consequences of the J-electronconfiguration. Many everyday observations in transition-metal chemistry have moreto do with the relative rates of reactions rather than the position of a thermodynamicequilibrium. So now we consider some of the kinetic manifestations of partiallyfilled d orbitals.

9.5.1 Kinetically Inert and Labile Complexes

It is convenient to divide the discussion of the mechanistic behaviour of transition-

metal complexes into those of labile and non-labile complexes, imperfect thoughthis division may be. The description of a complex as labile or non-labile isempirical, being based upon the typical time it takes for a reaction to proceed tocompletion. We adopt the suggestion of Taube, which refers to substitution reactionsin which one of the ligands coordinated to a metal center is replaced by anotherligand. If this process is complete in less than one minute (at 298 K with reactantconcentrations of 0.1 M), then the complex is described as labile, whereas if ittakes considerably longer than this time, the complex is described as non-labile orinert. In what follows, we should note two generalities. Firstly, inert complexes arenot necessarily thermodynamically stable with respect to the reaction underconsideration; conversely, thermodynamically stable complexes often undergo rapidreactions. Secondly, the properties of lability and inertness are found to be looselyassociated with particular metal ions in particular oxidation states; complexes ofcobalt(m), chromium(m) and most second and third row transition metals aregenerally inert.

9.5.2 Ligand Substitution Reactions

One of the commonest reactions in the chemistry of transition-metal complexes is

the replacement of one ligand by another ligand (Fig. 9-3) - a so-called substitutionreaction. These reactions proceed at a variety of rates, the half-lives of which mayvary from several days for complexes of rhodium(m) or cobalt(m) to about amicrosecond with complexes of titanium(m).

L- ^ ~L

L L

Figure 9-3. The substitution of L by X in an octahedral complex.

9.5 Consequences of the d-Electron Configuration upon Reaction Rates 187

The precise mechanism by which this process occurs has been the subject ofconsiderable study and debate over the past thirty years. Limiting mechanisms wouldinvolve an associative process in which an intermediate or transition state ofincreased coordination number is formed (the SN2 mechanism of organic chemistryrepresents a limiting associative process), or a dissociative one with an intermediateor transition state of lower coordination number (the limiting SN1 mechanism oforganic chemistry). These mechanisms differ in the relative importance of bond-making to the incoming ligand and bond-breaking with the leaving ligand in thetransition state. In general, the mechanisms are thought to be of the interchangetype in which bond-making with the incoming group is concurrent with bond-breaking to the leaving group. These reactions are delineated /a or /d dependingwhether bond-making to the incoming ligand or bond-breaking to the leaving ligandis considered to be dominant in the transition state. In many cases, conventionalkinetic studies do not provide data to allow unambiguous assignment of themechanism for substitution reactions. The reader is referred to the reading list at theend of this chapter for further information upon this vexing subject! As already mentioned, complexes of chromium(m), cobalt(m), rhodium(m) andiridium(in) are particularly inert, with substitution reactions often taking many hoursor days under relatively forcing conditions. The majority of kinetic studies on thereactions of transition-metal complexes have been performed on complexes of thesemetal ions. This is for two reasons. Firstly, the rates of reactions are comparable tothose in organic chemistry, and the techniques which have been developed for theinvestigation of such reactions are readily available and appropriate. The time scalesof minutes to days are compatible with relatively slow spectroscopic techniques.The second reason is associated with the kinetic inertness of the products. If theproducts are non-labile, valuable stereochemical information about the course ofthe substitution reaction may be obtained. Much is known about the stereochemistryof ligand substitution reactions of cobalt(m) complexes, from which certaininferences about the nature of the intermediates or transition states involved may bedrawn. This is also the case for substitution reactions of square-planar ds complexesof platinum(n), where study has led to the development of rules to predict thestereochemical course of reactions at this centre. It will not have escaped the reader's attention that the kinetically inert complexesare those of d3 (chromium(m)) or low-spin d6 (cobalt(m), rhodium(m) or iridium(m)).Attempts to rationalize this have been made in terms of ligand-field effects, as wenow discuss. Note, however, that remarkably little is known about the nature of thetransition state for most substitution reactions. Fortunately, the outcome of theapproach we summarize is unchanged whether the mechanism is associative ordissociative. Basolo and Pearson, in their classical work on inorganic reaction mechanisms,developed a ligand-field based approach to understanding the occurrence ofkinetically inert transition-metal ions. They calculated the LFSE associated with thestarting (octahedral) complex for a given dn complex (Dq values were deducedfrom actual spectroscopic data). They then considered limiting associative anddisocciative mechanisms leading to seven- or five-coordinate intermediates. Thefive-coordinate intermediate in the dissociative process might exist as a square-188 9 Chemical Consequences of the d-Electron Configuration

based pyramid or as a trigonal bipyramid. They attempted then to determine the

ligand-field stabilization energy for the intermediate by assuming that the overallmagnitude of the ligand-field from the ligands is unchanged on passing from thestarting complex to the intermediate. Although the justification for this may seemtenuous, it being argued that the change in number of ligands is countered bychanges in bond lengths, recent and detailed ligand-field analyses support this earlyidea. For example, the reduction in the ligand-field expected on going from six tofive ligands is balanced by the five ligands being closer to the metal ion in responseto the requirements of the electroneutrality principle. The ligand-field splittings fora variety of geometries were so calculated (in terms of Aoct). The change in LFSEbetween starting complex and intermediate was termed the ligand-field activationenergy (LFAE). A decrease in ligand-field stabilization energy upon passing fromthe ground state to the transition state (a positive LFAE in their definition), wouldprovide an additional contribution to the overall activation energy for the substitutionprocess. A negative contribution corresponds to a lowering of the activation energyfor substitution. Basolo and Pearson found that the LFAE for substitution of d3 orlow-spin d6 was positive regardless of the coordination number or geometry of thetransition state. [Perhaps this is just another way of stating that the LFSE foroctahedral d3 or low-spin d6 centers is high!] Clearly, the approach hinges upon thereliability of the estimates of the ligand-field stabilization energy for the transitionstate, whose detailed geometry is unknown.

Box 9-2

Ociahedron Square-based Pentagonal

pyramid bipyramid

The diagram shows the splitting diagrams which are used in calculations of this type. The LFSE for an octahedral d3 ion is-1.2^oct. The estimated LFSE's for square-planar five coordinate and pentagonal bipyramidal seven-coordinate transition states are -1.04>ct and -0.774z\oct respectively, leading to LFAE's of +0.2Z\ oct and +0.426z\oct respectively. 9.5 Consequences of the d-Electron Configuration upon Reaction Rates 189

9.5.3 Rates of Electron Transfer Reactions

In the same way that we considered two limiting extremes for ligand substitutionreactions, so may we distinguish two types of reaction pathway for electron transfer(or redox) reactions, as first put forth by Taube. For redox reactions, the distinctionbetween the two mechanisms is more clearly defined, there being no continuum ofreactions which follow pathways intermediate between the extremes. In one pathway,there is no covalently linked intermediate and the electron just "hops" from onecenter to the next. This is described as the outer-sphere mechanism (Fig. 9-4).

2+ Z+ π 3+ 3+ _ -, L L Ί

L L L L

M1(II) M2(III) M1(III) M2(II)

Figure 9-4. The outer-sphere mechanism for an electron transfer reaction betweentwo complexes. No covalently-linked intermediate is involved in the reaction.

The second mechanism involves the formation of a covalent bridge through whichthe electron is passed in the electron transfer process. This is known as the inner-sphere mechanism (Fig. 9-5). The inner-sphere mechanism is restricted to those complexes containing at leastone ligand which can bridge between two metal centers. The commonest examplesof such ligands are the halides, hydroxy or oxo groups, amido groups, thiocyanate

3+ 5+

I ^L L^ | L L L L

M1(II) M2(III) Mj(II) M2(III)

3+ 2+ 5+ L

L L 4'· L r L L' L I ^i

M1(III) M2(II) M1(III) M2(II)

Figure 9-5. The inner-sphere mechanism for an electron transfer reaction between twocomplexes. A covalently-linked intermediate is involved in this reaction.190 9 Chemical Consequences of the d-Electron Configuration

and more complex conjugated organic ligands such as pyrazine (9.12) or 4,4'-bipyridine (9.13).

.N

C 9.12

The scheme in Fig. 9-5 above illustrates the case in which the bridging ligand,X, is transferred from metal center M1 to M2 in the course of the reaction. Althoughthis is not a necessary consequence of an inner-sphere pathway, it is often observed,and provides one method for establishing the mechanism. It is often very difficult to distinguish one mechanism from another, but someclever experiments based upon LFSE effects have been designed. In the previoussection, we established that d3 and low-spin d6 metal complexes are kineticallyinert, and only undergo ligand substitution and displacement reactions with difficulty.Study of electron transfer reactions between two such kinetically inert metal centerssuggests that the redox processes proceed by outer-sphere mechanisms, since wecannot form the new metal-ligand bonds necessary in forming the bridged inter-mediate for an inner-sphere mechanism. Typical studies have involved cobalt(m),chromium(in), iron(n), ruthenium(n) and osmium(n) complexes. A typical exampleof a reaction involving two kinetically inert reactants is given in Eq. (9.25).

[Ru(bpy)3]2+ + [Co(Cn)3J3+^ [Ru(bpy)3]3+ + [Co(en)3]2+

d6 d6 d5 d1 (9.25) inert inert labile labile.

In the case of other systems in which one or both of the reactants is labile, nosuch generalization can be made. The rates of these reactions are uninformative,and rate constants for outer-sphere reactions range from 10~9 to 1010 sec~!. Noinformation about mechanism is directly obtained from the rate constant or the rateequation. If the reaction involves two inert centers, and there is no evidence for thetransfer of ligands in the redox reaction, it is probably an outer-sphere process. However, some quantitative interpretation of the rates of outer-sphere reactionsmay be made. It is possible to determine the rate constant, kn for the reaction oftwo complex ions [M1L6J2+ and [M2L6J3+ (Eq. 9.26).

[M1L6J2+ + [M2L6J3+ -> [M1L6J3+ + [M2L6J2+ Jk 12 (9.26)

Marcus and Hush have developed a theory, which bears their names, that relates thevalue of k12 to the rates (kn and k22) of the 'self-exchange' reactions of the two 9.5 Consequences of the d-Electron Configuration upon Reaction Rates 191

components (Eq. 9.27 and 9.28) and the stability constant Kn for the reaction ofinterest.

[M1L6J2+ + [M1L6]3+ -> [M1L6J3+ + [M1L6J2+ Jk 11 (9.27)

[M2L6J2+ + [M2L6J3+ -» [M2L6J3+ + [M2L6J2+ k22 (9.28)

In many cases, the values of kn and k22may be directly or indirectly determined.

We shall say no more about this relationship here, other than to indicate that itproves to be generally applicable, and is sufficiently accepted that the Marcus-Hushequation is now used to establish when an outer-sphere pathway is operative. In thecontext of this chapter, the involvement of the Ku term is interesting for it relatesto the relative stabilization of various oxidation states by particular ligand sets. Thefactors which stabilize or destabilize particular oxidation states continue to playtheir roles in determining the value of Kn, and hence the rate of the electron transferreaction. There is a very special case for self-exchange reactions in which the left side ofthe equation is identical to the right side. Accordingly, there is no free energychange in the reaction, and the equilibrium constant (Kn) must be unity (Eq. 9.29).

[Co(NH3)6]3+ + [Co(NH3)6]2+ ^ [Co(NH3)6]2+ [Co(NH3)6]3+ (9.29)

AG0 = 0 IgA: = O

However, metal ions in higher oxidation states are generally smaller than thesame metal ion in lower oxidation states. In the above example, the Co(Ii)-N bondsare longer than Co(m)-N bonds. Consider what happens as the two reactants cometogether in their ground states and an outer-sphere electron transfer occurs. Weexpect the rate of electron transfer from one center to another to be very muchfaster than the rate of any nuclear motion. In other words, electron transfer is verymuch faster than any molecular vibrations, and the nuclei are essentially staticduring the electron transfer process (Fig. 9-6). Thus, the interaction of the ground state cobalt(n) complex with long Co-Nbonds and the ground state cobalt(m) complex with shorter Co-N bonds initially

2+ 3+ I 3+ 2+ L L^ I ^L L^^ ^L Co L^^ I ^^L ^cor ^^ L Vr ^^ ^L - L

Co(III) Co(U) Co(II)* Co(III)*

Figure 9-6. The consequences of a self-exchange electron transfer between a ground statecobalt(n) and a ground state cobalt(m) complex.192 9 Chemical Consequences of the d-Electron Configuration

occurs without any rearrangement of Co-N bond lengths. Both products will, ofcourse, be in vibrationally excited states. The cobalt(n) complex will havecompressed Co-N bonds whilst the cobalt(m) complex will have extended Co-Nbonds. At a later stage, these must relax to the equilibrium Co-N bond lengthsappropriate for each oxidation state. This step involves the emission of energy. Yetthere is no overall energy change in the reaction. The requisite balance derives froman activation energy associated with the electron transfer process. In order for thereto be no overall energetic change in the electron transfer self-exchange reaction, theelectron transfer must occur between vibrationally excited species with equivalentbond lengths (Fig. 9-7).

Co(II) Co(III) Co(II)* Co(III)*

This is the origin of the various values for self-exchange rate constants. We maynow attempt to rationalize some of these in terms of the ^/-electron configurationsof the various oxidation states. Consider the self-exchange rate constants for someiron complexes.

[Fe(phen)3]2+/3+ k = 1038 M-11 s-11

[Fe(CN)6]4-73- k = io M-1 s-1 [Fe(H20)6]2+/3+ k = 10 M- s-

The water complexes are high spin, whereas the cyanide and phen complexes arelow spin. In the case of the cyanide and phen complexes, the interconversion of thet2giron(ii) and t2giron(m) states simply involves the loss or gain of an electron fromthe t2g level. Since these are the orbitals oriented between the ligand donor atoms, 9.5 Consequences of the d-Electron Configuration upon Reaction Rates 193

there will only be minimal changes in the electron-ligand repulsions and smallconsequent changes in the Fe-ligand distances. The small changes in Fe-liganddistances mean that the activation energy for the electron transfer reaction will below, and the rate of the reaction will be high. In the water complex, the t^e\ iron(n)and i2g£g iron(m) states are involved. Once again, the electronic changes are occurringin the t2g manifold. It is thought that the reason for the very rapid reactions of thephen complexes are due to the involvement of the π* orbitals of the ligand in theelectron transfer process. An interesting contrast is seen when we consider related reactions involvingcobalt (Eq. 9.30). In this case, there is a spin state change in the electron transferprocess. This results in two separate contributions to the high activation energy forthe self-exchange. The high-spin cobalt(n) complex possesses two electrons in theeg orbitals. These are oriented directly towards the ligands, and electron-ligandinteractions are expected to result in long Co-ligand distances. In these complexes,the Co(Ii)-N distance is 2.11 A and the Co(m)-N distance is 1.93 A. The activationenergy is high because of this difference in bond lengths, but also because of theelectronic rearrangement that is needed in the process. No longer do we simplymove an electron from one center to another for now a rearrangement of electronsis to be achieved. For the same reasons underlying the Franck-Condon principle,we expect to have electron transfer between electronically excited states (Eqs. 9.31or 9.32). Further discussion of this topic is beyond the scope of this book.

[Co(NH3)6]3+ + e- ^ [Co(NH3)6]2+ k = IQ-9

low spin high spin (9.30)

Co(IIi) -»Co(m)* ->Co(n) (9.31)

f 6 f 5pl f 5~2 /2g hgtg i2gC g

Co(IIi) ->Co(n)* ->Co(n) (9.32)

f 6

We conclude with a consideration of a few other cobalt self-exchange reactions.

The reaction in Eq. (9.33) is faster than that involving the ammine complexes (Eq.9.30) because the water is a weaker-field ligand than ammonia. Thus, the activationenergy for the formation of the electronically excited states is lower, as is thechange in Co -ligand distances in the two oxidation states.

[Co(H2O)6I3+ + e- - [Co(H2O)6J2+ k = 1 M'1 s-1 (9.33)

low spin high spin

The reaction in Eq. (9.34) is also faster because the bpy ligand is a strong fieldligand and there is no longer any need for electronic rearrangement upon change inoxidation state. The process is now comparable to those discussed earlier for lowspin iron complexes.194 9 Chemical Consequences of the d-Electron Configuration

[Co(bpy)3]3+ + e- ^ [Co(bpy)3]2+ ^ = I M ' 1 s~l

the inner-sphere process in which a bridge is formed between the two metal centers.The J-electron configurations of the metal ions involved have a number of profoundconsequences for this reaction, both for the mechanism itself and for ourinvestigation of the reaction. The key step involves the formation of a complex inwhich a ligand bridges the two metal centers involved in the redox process. For thisto be a low energy process, at least one of the metal centers must be labile. A number of ingenious experiments have been devised to establish the operationof this mechanism. These all revolve about the lability or inertness of particular d-electron configurations. Remember that complexes of first row transition-metal ionswhich possess d3 or low-spin d6 electronic configurations are usually particularlyinert with respect to ligand substitution reactions - they are kinetically stabilized.Consider the reaction of the cobalt(m) complex [Co(NH3)5Cl]2+ with the chromium(n)complex [Cr(H2O)6J2+ in Eq. (9.35). The first step in an inner-sphere process wouldbe the formation of a chloro-bridged complex. The chloride ligand is better suitedfor bridging than either of the neutral water or ammonia ligands. The d6 cobalt(m)center is kinetically inert, but the d4 chromium(n) complex is labile. The intermediatebridged complex is thus formed by the displacement of a water molecule from thechromium, and with retention of the cobalt - chloride bonding. If this complexcollapses without electron transfer, it will be the labile Cr - Cl bond which breaks toregenerate the starting complexes.

If this complex now collapses, it will be the labile Co-Cl bond which is broken,as opposed to the inert Cr-Cl bond. The labile cobalt(n) complex reacts furtherwith bulk water to generate [Co(H2O)6J2+ (Eq. 9.37). The key feature is that anecessary consequence of this inner- sphere reaction is the transfer of the bridgingligand from one center to the other. This is not a necessary consequence of all suchreactions, but is a result of our choosing a pair of reactants which each changebetween inert and labile configurations. In the reaction described above, the chloride 9.5 Consequences of the d-Electron Configuration upon Reaction Rates 195